Magnetism

MAGNETISM

THE word magnetism is derived from the Greek word μ,ayvη<s, which was applied to an ore of iron possessing a remarkable attractive power for iron, and supposed to have been originally found near the town of Magnesia, in Lydia. Thus Lucretius writes:—

Quern Magneta vocant patrio de nomine Graii, Magnetum quia fit patriis in finibus ortus.

This name is said by Plato to have been given to it by Euripides, and he adds that most call it the Heraclean stone. It is needless here to criticize the above or other derivations that have been given for the word; we merely remark that it is now applied to all the phenomena kindred to that which first drew attention to the magnetic iron ore, viz., a selective attraction for iron.

In the following article we shall give, in the first place, a sketch of the leading phenomena of strongly magnetic bodies. We shall then describe a provisional theory sufficient to render a general account of these phenomena, and shall afterwards proceed to render this theory more precise, to develop it to its necessary conclusions, and to compare these with experiment, indicating where the theory is either incorrect or incomplete. Then we shall discuss the paramagnetic and diamagnetic properties of all bodies, as expounded by Faraday; an account will be given of the connexion between the magnetic and the other physical properties of bodies; and, lastly, we shall endeavour to give some idea of the different physical theories that have been proposed in order to give something more than a mere shorthand record of the facts of observation.

Leading Phenomena.

It appears, from what Lucretius says in the passage above quoted, that the Greeks and Romans were aware, not only that the loadstone, or magnetic iron ore, attracted iron, but also that it endued iron in contact with it with its own peculiar property. Thus an iron ring will hang suspended by the attraction of a loadstone, and from that ring another, and so on, up to a certain number, depending on the power of the stone and the weight, &c., of the rings. They were also aware that the attraction was confined to iron, or at all events was not indiscriminate, and that it was not destroyed by the intervention of other bodies, such as brass, between the magnet and the iron. It appears, too, from the passage—

Fit quoque ut a lapide hoc ferri natura recedat

Interdum, fugere atque sequi consueta vicissim, &c.— that they had an idea that, under certain circumstances, the attraction might be replaced by a repulsion. If, however, we understand aright the latter part of Lucretius’s somewhat obscure description of what seems to have been an actual experiment of his own, this notion was in reality a hasty generalization, not justified by the observed facts. In any case there seems no warrant for assuming, as some have done, that the ancients had any definite conception of magnetic polarity.

What they wanted in definite experimental knowledge they supplied by an abundant use of the imagination.

We are told, for instance, that the magnet attracts wood and flesh, which was certainly beyond their powers of observation; that it is effective in the cure of disease; that it affects the brain, causing melancholy; that it acts as a love philtre; that it may be used in testing the chastity of a woman; that it loses its power when rubbed with garlic, but recovers it when treated with goat’s blood; that it will not attract iron in the presence of a diamond, and much else that was eagerly copied by the wonder-loving writers of the Middle Ages.

The science of magnetism made no real progress till the invention of the mariner’s compass. The early history of this instrument is very obscure. According to some authorities it was invented in China, and found its way into Europe probably through Arabian sources. The light thrown by recent researches on the literature of the Chinese has apparently thrown doubt upon their claim to this invention, although the knowledge of the loadstone and its attractive property may have been older among them than even among the Greeks. The first accounts of the compass in Europe go back to the 12th century, and, although the instrument described is very rough, it is not spoken of as a new invention. In its earliest form it seems to have consisted simply of an iron needle which was touched with the loadstone and placed upon a pivot, or floated on water, so that it could turn more or less freely. It was found that such a needle came to rest in a position pointing approximately north and south (some accounts say east and west, in which case there must have been a cross piece on the needle to indicate what was probably the important direction for the mariner). As these compasses were made of iron (steel was not used till much later), and were probably ill-pivoted, they must have been very inaccurate; and the difficulty of using them must have been much increased by the want of a card, which was a later addition made apparently by the Dutch.

It is unnecessary to enter into more detail here respecting the early history of the compass, as the matter has been very fully treated in the article Compass. We proceed therefore to show the bearing of the invention upon the science of magnetism. It will at once be seen that it involves two scientific discoveries of capital importance :— first, that the loadstone can transmit to iron with which it comes in contact a permanent property like its own; and, secondly, that a loadstone or magnet if suspended freely will turn so that a certain direction in it assumes a fixed position relative to the geographical meridian, a certain part of the magnet turning always towards the north, and the part opposite towards the south. These opposite parts of the magnet are called its “poles.”

To fix our ideas we shall describe a process by which we might definitely determine this direction in the magnet. Following· the example of Gilbert, let us consider a spherical magnet. Our reason for dealing with this form in the first instance is to make it perfectly clear that the phenomena depend essentially on something apart from the form of the body. We shall suppose that the magnet is homogeneous as to its mass, so that its centre of gravity coincides with its centre of figure. Suspend this spherical magnet by a fine thread of untwisted silk, attached to any point of its surface, say P. After the magnet has come to rest, mark the vertical plane through the centre which falls in the geographical meridian; this may be done by tracing a great circle on the surface of the magnet. Next find the point P' in which the vertical through P cuts the surface again, and suspend the magnet by P', again marking the plane which falls in the meridian. Now, find the plane which bisects the acute angle between the two former planes, mark it by a great circle, and call it the axial plane of P. If we thus find the axial planes of any number of points, we shall find that they all intersect in one common line passing through the centre of the sphere. We may call this line the “axis” of the magnet. Let us mark the points where it cuts the surface; we may call these the “poles” of the magnet. We shall then observe that, however we suspend it, the magnet will always come to rest so that the vertical plane through the axis makes a definite angle with the meridian. This angle (δ) is called the “declination” (also, by sailors, the “variation”.); it varies from place to place, and from time to time, but very slowly, so that throughout a limited area of the earth’s surface, and for a limited time, it may be regarded as constant.

One end of the axis always turns northwards, and the other always southwards; we shall call the former the “north” and the latter the “south pole,” although, for reasons to be afterwards explained, it would be more appropriate to invert the order of these names. Henceforth the vertical plane in which the axis of the magnet comes to rest will be called the magnetic meridian, and the two horizontal directions in this plane magnetic north and magnetic south respectively.

It must be carefully noticed that there is a certain amount of arbitrariness in our definition of the axis and poles of a magnet. In reality it is only the direction of the axis that is fixed in the body, and not its absolute position. This will be made plain if we repeat all our experiments with the spherical magnet after fastening to it a piece of wax or other non-magnetic body, so as to leave its magnetic properties unchanged, but to throw its centre of gravity out of the centre of figure. Everything will fall out as before, only the axial planes of the different points of suspension will now meet in a line, parallel, it is true, to the axis determined before, but passing through the new centre of gravity. In point of fact, therefore, we might choose any point in the body, draw a line through it iu the proper direction, and call this the axis. Hereafter we shall, unless the contrary is stated, draw the axis through the centre of gravity of the body, or through its centre of figure if it has one; and we define the poles, for the present, as the points in which the axis cuts the surface of the magnet, supposing, as will be generally the case, that the line cuts the surface iu two points and no more.

Having now obtained a definite idea of the axis of a magnet, and seen that it has, in the first instance at least, nothing to do with the external form of the body, let us proceed to make an artificial magnet of the particular kind usually called a “magnetic needle,” and briefly examine its properties. Take a tolerably thin flat piece of pretty hard-tempered steel, of the elongated symmetrical form NS shown in fig. 1. We suppose it, in the first place, in an unmagnetic condition. Let it be pierced by a well-turned axis ab, passing accurately through its centre of gravity, and perpendicular to its plane, so that, when the axis is placed on two horizontal knife edges, the needle will rest in any position indifferently. Further, let four very small hooks, c, d, e,f, be attached, two (c, d) to the ends of the axis, and other two (e,∕) to the edges of the needle in a line perpendicular to NS. Now rub the half of the needle towards N with the south pole of the spherical magnet whose properties we have just discussed, beginning the stroke at the middle and ending it at the point of the needle, and for symmetry’s sake let us do the same to the other side of the needle, and then repeat this process with the north pole of the sphere on the other half towards S. Let us examine the properties of the needle thus “magnetized.” If we suspend it first by the hook c and then by the hook d, we shall find that in both cases the line joining NS makes very nearly the same angle with the geographical meridian. Hence the magnetic axis must lie in a plane through NS perpendicular to the plane of the needle. A similar experiment with the two hooks e, f will show that the magnetic axis lies approximately in the plane of the strip, which we may suppose for the present to be infinitely thin. Hence the magnetic axis may be taken to be coincident with the line NS joining the points of the needle. This coincidence is, however, in general only approximate, and in delicate measurements corrections have to be made on that account, of which more hereafter. If we now mount our magnetized needle on a piece of cork or two straws, and float it in a basin of water, or replace its axle by a small cap and set it on a pivot, we have the mariner’s compass in its early form. We shall call it a magnetic needle, to distinguish it from the more elaborate compass of the present day. A favourite way of showing the directive property of a magnet, described by Gilbert, is to magnetize a sewingneedle, and lay it very gently, by means of a fork of wire, on the surface of water; it will float and turn until it takes up its position in the magnetic meridian.

A needle mounted in this way, so as to have great freedom to move in a horizontal plane, is of great use in magnetic experiments. Gilbert calls it a “versorium.” When very delicate applications are in view, the point of the pivot on which it is mounted must be very hard (say of hard tempered steel or iridium), and the cap should be fitted with an agate or other hard stone having a polished cavity of the form of a blunted cone to receive the pivot. A still better arrangement, also used by Gilbert, is to suspend a short and very light piece of steel wire—a fine sewing needle may be used— by means of a single fibre of silk. The most delicate arrangement of all is to use one of Sir W. Thomson’s light galvanometer mirrors with the magnets attached, and follow its movements by means of the lamp and scale as usual. See Galvanometer.

Such, with as much of modem accuracy imported into them as was necessary for clearness of exposition, were the facts of magnetism as known up to the beginning of the 16th century.

Another experiment with our magnetized needle will enable us to describe the next important magnetic dis covery. In its unmagnetized condition the needle rested indifferently in any position when its axis was placed on two horizontal knife edges. In the magnetized state this is no longer the case. The axis of the needle now takes up a fixed position, with its north end pointing downwards (fig. 2), and if disturbed will oscillate about that position, and finally settle into it again. The angle which the axis NS makes with the horizon is least when the plane of rotation of the needle is in the magnetic meridian: the angle (t) in this case is called the “dip,” or (by Continental writers) the “inclination.” It is greatest, viz., 90°, when the plane of rotation of the needle is vertical and perpendicular to the magnetic meridian. At Greenwich the dip is about 67° 30' at the present time. If we place the needle with its plane of rotation perpendicular to the line of dip, the equilibrium will be indifferent, as it was in all positions before magnetization; but there is no other position of the magnetized needle for which this is true.

The remarks which we made as to variation in space and time of the declination apply also to the dip. The variation from place to place differs, however, in nature from that of the declination. Along a line running in the neighbourhood of the geographical equator, partly north and partly south of it, the dip is zero. North of this line, which is called the magnetic equator, the north end of the needle dips below the horizon; and the angle of dip increases as we go northwards, until, at a point in the Hudson’s Bay Territory, the needle dips with its north pole vertically downwards. South of the magnetic equator the south end dips below the horizon; and there is again a point in the southern hemisphere where the south end dips vertically downwards. These points are called the “magnetic poles” of the earth. For further details on this subject we refer the reader to the discussion of terrestrial magnetism in the article Meteorology.

It was in the accurate observation of the declination and dip of the magnetic needle that the science of magnetism arose. The dip appears to have been first observed by Georg Hartmann, vicar of the church of St Sebaldus at Nuremberg (1489-1564), who seems to have been in advance of his age in magnetical matters. In a letter to Duke Albrecht of Prussia, dated 4th March 1544, he writes :—

“Besides, I find this also in the magnet, that it not only turns from the north and deflects to the east about 9° more or less, as I have reported, but it points downwards. This may be proved as follows. I make a needle, a finger long, which stands horizontally on a pointed pivot, so that it nowhere inclines towards the earth, but stands horizontal on both sides. But as soon as I stroke one of the ends (with the loadstone), it matters not which end it be, then the needle no longer stands horizontal, but points downwards (fällt unter sich) sonie 9° more or less. The reason why this happens was I not able to indicate to his Royal Majesty.”

From this it will be seen that Hartmann had unquestionably observed the tendency of the magnetized needle to dip. His method of observing is of course unsuited for measurement, and it is not surprising that he got a result of 9° instead of somewhere about 70°.

In 1576 the dip was independently discovered by Robert Norman, a skilful seaman and an ingenious artificer, according to Gilbert. He was in the habit of making compass needles, and carefully balancing them so as to play Horizontally on their pivots before magnetization. He found that, after they were magnetized, they constantly dipped with the north end downwards, so that a counterpoise had to be added to bring them back to the horizon. This led him to construct a special instrument, the prototype of the modern dipping needle, to show this new phenomenon. With this instrument he made the first accurate measurement of the dip, and found it to be 71° 50' at London.

The early English magnetic observers, of whom Norman and Burroughs (who wrote an able supplement to Norman’s work) were admirable examples, must have done much for the introduction of precise ideas into magnetism. But their fame was speedily eclipsed by William Gilbert of Colchester (1540-1603), whom Poggendorff has justly called the Galileo of magnetism, and whom Galileo himself thought enviably great. In his great work entitled De Magnete Magneticisque Corporibus et de Magno Magnete Tellure Physiologia Nova, first published in 1600, we find a complete account of what was known of magnetic phenomena up to his time, with a large number of new ideas and new experimental facts added by himself. We find in Gilbert’s work, in a more or less accurate form, nearly all that we shall lay before the reader in the first section of this article, described very much in the language that we shall use. “How far he was ahead of his time is best proved by the works of those who wrote on magnetism during the first few decades after his death. They contributed in reality nothing to the extension of this branch of physical science.” Mutual Action of Like and Unlike Poles.— If we take a magnet whose poles N', S' have been determined and marked as above explained, and bring its north pole N' near the north pole N of a magnetic needle, N will move in a direction indicating repulsion between N and N'. The same result will follow if the south pole S' of the magnet be brought near the south pole S of the needle. But if S' be brought near N, or N' near S, attraction will be indicated. Hence the following fundamental law of the action between two magnets :— Like poles repel each other; unlike poles attract each other. It would appear, therefore, that the whole action of one magnet upon another is of a somewhat complicated character, even if we take the simplest view of it that the experimental facts will allow, viz., that the action may be represented by forces acting between the two pairs of points in each magnet which we have defined as north and south poles. On this assumption, the action of N'S' upon NS would consist of the four forces represented in fig. 3, for all these must exist in accordance with the law just established. Whether this is a sufficient representation of the most general case, and what the exact law of the forces ought to be, we are not yet in a position to decide. One thing, however, is clear, that the action between two poles must diminish when the distance between them increases; otherwise we should not have been able to make the action of N or S upon N' prevail, by bringing the one or the other nearer.

It was perhaps the complexity of this analysis (along with the fact that the action of the magnet upon soft iron, which was the earliest discovered magnetic phenomenon, is not a pure case of this action, but involves also another phenomenon, viz., magnetic induction) that prevented for so long the discovery of the elementary law we are now discussing. At all events, it seems to have been a new discovery in the 16th century, if we may judge from a passage in the letter of Hartmann above alluded to. He was certainly aware of the existence of magnetic repulsion in some form or other. It is somewhat difficult to gather from his description what it was exactly that he observed, and he nowhere states the law fully and explicitly. In Norman’s Newe Attractive we find it clearly stated, and demonstrated by means of a needle floating on water or suspended by a thread; yet he does not appear to claim the fact as his discovery. If, therefore, Hartmann was not the actual discoverer, we may at least conclude that the law became familiar to magnetic philosophers during the thirty years that separated him from Norman.

The Magnetic Field. —We next introduce a method of conceiving and describing magnetic actions which was invented and much used by Faraday. Since a magnet acts upon a magnetic needle placed anywhere in the surrounding space, we call that space the magnetic field of the magι∣et Neglecting the earth’s magnetism, we may map out this field as follows. Conceive any plane drawn through the axis of the magnet, and place it so that this plane shall be horizontal. Then at any point in this plane place a very small magnetic needle, and note the direction which its axis assumes under the action of the magnet; then proceed to move the centre of the needle in the direction in which its north pole points, and continue the motion so that at each point the centre is following the direction indicated by the north pole. The line thus traced will at last cut the surface of the magnet at some point lying towards its south pole; and if we continue the line backwards, by following the direction continually indicated by the south pole of the needle, it will cut the surface of the magnet at some point lying towards the north pole. Such a line is called a line of magnetic force; and, since one such line can be drawn through every point of the plane, and any number of planes can be taken through the axis of the magnet, we can conceive the whole magnetic field filled with such lines. Fig. 4, taken from Faraday, gives an idea of the distribution of the lines of force in the field of a bar magnet; fig. 5 represents the lines in the field due to two neighbouring like poles.

These diagrams were not obtained by the method we have just described, but by a much simpler process which we shall describe by and by. Their use, so far as we have gone, is to tell us how a small needle, free to move about its centre in any direction, will place itself at any part of the field, viz., it will place its axis along the tangent to the line of force which passes through its centre, its north pole pointing in that direction which ultimately leads to the south pole of the magnet producing the field.

Suppose we apply these ideas to a spherical magnet (a terella, or earthkin, as Gilbert calls it). The lines of force in any plane through its axis would be found to run something like the curves in fig. 6. If, therefore, we carried a small needle (suspended from a silk fibre so as to be perfectly free to move in all directions) round the magnet in a meridian plane, its axis would constantly remain in the meridian plane, its north pole always point towards the south pole of the spherical magnet, but dip more and more below the tangent plane to the sphere as the centre recedes from the equator, and end by pointing straight towards the south pole when the centre reaches the magnetic axis (see fig. 6).

When we reflect that in all. our experiments the properties of magnets, whether native, such as the loadstone, or artificial, such as the needles magnetized by rubbing with the loadstone, have proved alike, and that every purely magnetic action on a magnet has its source in some other magnetic body, we are naturally led to the conclusion that the reason why at every point of the earth’s surface the axis of a freely suspended magnet assumes a definite position is simply that the earth itself is a great magnet, and that in observing the declination and dip we are simply exploring the magnetic field of the earth. It is true that, according to the experiment above described, the declination would every where be zero, and the magnetic equator would coincide with the geographical, but that arises merely because we assumed our earthkin, for simplicity of explanation, to be symmetrically magnetized, so that its lines of force ran in planes passing through its axis. It remains to be discussed whether the most general assumption, viz., that the earth is a magnetic body, will not account for the facts of terrestrial magnetism. The answer to this question has been given, as we shall see, by Gauss.

This idea, whose simplicity is the truest measure of its greatness, is due to Gilbert, and was by him made the foundation of his work on magnetism. The boldness of his theory will be appreciated when we remind the reader that in his day the dip was but newly discovered, and had been measured only at London, so that Gilbert’s very full and clear exposition of this phenomenon, which we have given above, was in fact a scientific prediction, which was not fully verified till long afterwards. Before Gilbert a variety of wild conjectures had been made as to the cause of the directive property of the magnet. Many, like Columbus, Cardan, and Paracelsus, believed that the magnet was attracted by a point in the heavens, possibly some magnetic star. Others supposed that the attracting point was situated in the earth; Fracastorius imagined hyperborean mountains of loadstone situated near but not quite at the north pole; and to this theory others contributed the detail that the magnetism of these mountains was so powerful that ships in these regions have to be built with wooden nails instead of iron ones, which would be instantly drawn out by the magnetic attraction.

It is clear that, if we call that magnetic pole of the earth which lies in the northern hemisphere its north pole, we ought, in accordance with our fundamental law of magnetic action, to call the northseeking pole of an ordinary magnet a south pole. When it is necessary to speak of magnets from this point of view, the difficulty is got over by calling the north-seeking pole the austral pole, and the south-seeking pole the boreal pole. In reality the danger of confusion is more imaginary than real. The reader should be warned, however, that in some French works the ordinary nomenclature is reversed, and that Faraday uses “marked” and “unmarked,” and Airy “red” and “blue,” in the sense in which north and south are commonly used.

The Earth's Action on a Magnet is a Couple.— Norman in his Newe Attractive (chapters v. and vi.) discusses very acutely the question whether there is any force of translation exerted upon a magnet. He advances three conclusive experiments to prove the negative. First, he weighed several small pieces of steel in a delicate gold balance, and then magnetized them, but could not detect the slightest alteration in their weight, “though every one of them had received vertue sufficient to lift up his fellow.” Secondly, he pushed a steel wire through a spherical piece of cork, and carefully pared the latter so that the whole sank to a certain depth in a vessel of water and remained there, taking up any position about the centre indifferently. After the wire was magnetized very carefully, without disturbing its position in the cork, it sank to the same depth as before, neither more nor less, the only difference being that now the wire set itself persistently in a definite fixed direction parallel to the magnetic meridian, the north end dipping about 71° or 72° below the horizon. Thirdly, he arranged a magnetized needle on a cork so as to float on the surface of water, and found that, although it set in the magnetic meridian, there was not the slightest tendency to translation in any direction. He concludes that there is no force of translation on the magnet, either vertical or horizontal He was evidently somewhat puzzled how to put this result into a positive form, and his “point respective,” as he calls it, is not a very clear explanation of the earth’s action. What he wanted was the modern idea of a “couple,” 7.e., a pair of equal but oppositely directed parallel forces acting on the two ends of the needle; but such an idea was not conceived in Norman’s day. Gilbert adopts Norman’s result in this matter, adding nothing essential, reproducing even Norman’s diagram of the spherical cork with the wire through it. It is clear therefore that Gilbert had a forerunner in the practice, as Bacon had in the theory, of inductive science; for Norman says, speaking of the mass of fables that had passed for truth in geography, hydrography, and navigation before his time, “I wish experience to bee the leader of Writers in those Artes, and reason their rule in setting it downe, that the followers bee not Jed by them into errors, as oftentimes have beene seene.”

The Magnetic Property is Molecular.— Apart altogether from the question as to how we are to represent the action of a magnet ιipon other magnets, there arises another quite distinct question, as to where the cause of this action resides. That these two questions are really distinct, although there has always been a tendency in the more superficial treatises on the subject to confuse them, will be obvious from the fact that we shall afterwards obtain more than one perfectly general way of representing the action of a magnet at external points, whereas there must be one and only one cause of this action. A very old experiment at once throws considerable light on this point. If we break a bar magnet into two pieces, it will be found that each of these is itself a magnet, its axis being in much the same direction as that of the original magnet, and its poles in corresponding positions, see fig. 7. The same holds if we break the bar into any number of pieces; and, quite generally, if we remove any piece, however small, from a magnet, this piece will be found to be magnetic, the direction of its axis usually bearing a distinct and easily recognizable relation to the direction of the axis of the whole magnet. We are therefore driven to the conclusion that the magnetic quality of a body is related to its ultimate structure, and not simply to its mass as a whole, or to its surface alone; and this conclusion is not to be invalidated by the fact that we can in general, as will afterwards appear, represent the action of the magnet at external points by means of a proper distribution of centres of attractive and repulsive forces upon its surface merely.

Temporary Magnetism of Soft Iron and Steel in the Magnetic Field.— Bodies which possess permanent magnetic properties, not depending on the circumstances in which they are placed, we shall henceforth call “permanent magnets.” The law of the action of one permanent magnet upon another, as we have seen, is that like poles repel and unlike poles attract each other. The action of a permanent magnet on pieces of soft iron is, at first sight, different, for either pole attracts them alike.

To fix our ideas let us take a small thin bar of soft iron or of steel, and test it with a delicate magnetic needle. It will usually be found, more particularly if a steel bar is taken, that one end of the bar will repel one or other of the poles of the needle. This is a sure sign of permanent magnetism. If, however, we heat the bar to whiteness and allow it to cool in a position perpendicular to the earth’s magnetic force, all permanent magnetism will be found to have disappeared. If we now place the bar in a horizontal plane (fig. 8) with its axis perpendicular to the axis of the needle, and one of its ends A, B near either pole of the needle, that pole will be attracted, no matter whether it be the north pole or the south pole of the needle, or which end of the bar be used.

Care must be taken in this experiment to avoid using a too strongly magnetized needle, and to keep the needle from touching the bar, otherwise the bar may receive traces of permanent magnetism which will disturb the result. It is very easy, by repeating the above experiment with an unmagnetized needle, to show that the power that the bar acquires of attracting the poles of the needle is temporary and depends on the presence of a magnetized body.

Keeping to our principle that a magnetic cause is to be sought for every magnetic action, we are led to explain the above experiment by saying that in the magnetic field a bar of soft iron or of unmagnetized steel becomes magnetic in such a way that its north pole points as nearly as may be in the positive direction of the lines of force passing in its neighbourhood (or, in other words, in the direction, as nearly as may be, in which a magnetic needle would point if placed in its neighbourhood). A body which becomes magnetic iu this way by the magnetic action of another body is said to be “magnetized” by “induction.” We shall suppose, in the meantime, that it loses all the magnetism thus acquired when the inducing action is withdrawn; although this is not necessarily,· and in fact not generally, the case, as we shall see by and by. The reason why soft iron is attracted by a permanent magnet is therefore now said to be that the iron becomes magnetic by induction, and is then acted upon by the magnet like any other magnet similarly placed. The accuracy of this analysis of the phenomenon may be confirmed by many simple but striking experiments, such as the following.

In the experiment above described, instead of placing the non-magnetic bar in a horizontal plane, place it in the plane of the magnetic meridian with its axis in the direction of the earth’s force (i.e., parallel to the line of dip). The lower end of the bar will then be found to repel and the upper end to attract the north pole of the needle (figs. 9, 10). This is at once explained on the above hypothesis; for the bar will be magnetized inductively by the earth’s force, so that its lower end becomes a north pole, and its upper end a south pole.

Let NS (fig. 11) be a bar magnet placed horizontally so that its axis produced passes through O, the centre of suspension of the needle sn, then the needle will be deflected in the direction of the arrow. If now we place between S and O a small sphere of soft iron, this deflexion will be increased, the reason being that the sphere is magnetized by induction, having a south pole towards O and a north pole towards S, and the action of these is added to that of NS.

Let NS (fig. 12) be a magnet placed in the magnetic meridian, ris' a small magnetic needle in the same horizontal plane, with its centre in the line bisecting NS at right angles. When acted on by NS alone, n's' will place itself parallel to NS, with its north pole pointing in the direction NS

Let the dotted line represent a line of force. If we move a small piece of soft iron ns along this line in the direction from N towards S, it will first deflect the needle as in fig. 13, and finally as in fig. 14, and in each position reversing it end for end will not alter the effect. All this is at once explained by the above hypothesis.

A variation of the last experiment may be made thus. Place a magnet vertically, in the neighbourhood of a magnetic needle; by moving it up and down a position will be found in which the action of the magnet on the needle is wholly vertical, so that the needle is not deflected from the magnetic meridian. Now take a small piece of soft iron and move it along a line of force passing near the needle, proceeding from the north to the south pole of the vertical magnet. It will then be found, in accordance with our hypothesis, that the north pole of the needle is first repelled, and finally attracted by the soft iron.

If we hang two short pieces of iron wire alongside of each other by parallel threads, they will be found to repel one another, and to hang separated by a considerable interval when a magnet is brought under them (see figs. 15 and 16). This experiment is due to Gilbert, who rightly explains it by saying that the two ends nearer the magnetic pole S become like poles of opposite kind to S, while the two farther ends are like poles of the same kind as S. The experiment may be varied by placing some little distance below the pole of a magnet S a piece of mica or thin card board M, and placing below that a short piece of soft iron wire; it will remain adhering to the mica, and so long as it is alone will hang more or less nearly vertical, but when another is placed alongside of it the two will diverge as in fig. 17.

One of the most interesting examples of magnetic induction is furnished by the action of a magnet on iron filings. If we plunge a magnet into a quantity of iron filings and then remove it, we find it thickly fringed around the poles, where the filings adhere to the magnet and to one another so as to form short bushy filaments: the thickness of the fringe diminishes very rapidly towards the middle of the magnet, where very few adhere at all. These filaments are composed of magnetized particles of iron adhering by their unlike poles.

If we place a small bar magnet under a piece of moderately rough drawing paper, strewn as uniformly as possible with fine iron filings, and then tap the paper very gently so as to relieve the friction, and allow each filing to follow tho magnetic action, then the filings will be seen to arrange themselves in a series of lines, passing, roughly speaking, from pole to pole, as in fig. 4 (p. 222). The explanation of this phenomenon is simply that each filing becomes magnetized by induction, and, if it were quite free to move about its centre, it would not be in equilibrium until it set its longest dimension along the line of force through its centre. The roughness of the paper effectually prevents translation, but does not hinder rotation, especially when the friction is relieved by tapping; hence every filing does actually set as if it were a little magnetic needle, subject of course to some slight disturbance from the neighbouring filings. The whole therefore assumes a grained structure, and the graining runs in the direction of the lines of force. We have thus an extremely convenient way of representing these lines to the eye, which lends itself in a variety of ways to the illustration of magnetic phenomena. In fig. 5 are shown the lines formed in the field near two like magnetic poles. These magnetic figures may be fixed in a great variety of ways, and projected on a screen so as to be visible to a large audience, but it is scarcely necessary to dwell here upon details of this kind.

These magnetic curves seem to have fixed the attention of natural philosophers at a very early period. They were originally called the magnetic currents, from an idea that they represented the stream lines of magnetic matter, which explained the magnetic action according to the theory then in vogue. La Hire mentions them, Mem. de ΓAcad, i 1717. Bazin gives an elaborate account of them in his Description des Couraηs Magnétiques dessines djaprès Hαture, Strasburg, 1753. Musschenbroek seems to have been the first to give the correct explanation depending on magnetic induction, Diss, de Magnete, 1729.

If the filings be laid very thickly on the paper, and one pole of the magnet be brought under them at a short distance off, they will arrange themselves in a pattern, and at the same time bristle up so as to stand more or less erect, according as they are nearer or farther from the magnet. They have thus the appearance of being repelled from the magnet. It was, in all probability, this phenomenon that was observed by Lucretius when he says (vi. 1042):—

“Exultare etiam Samothracia ferrea vidi, Ac ramenta simul ferri furere intus alienis In scaphiis, lapis hic Magnes cum subditus esset.”

His conclusion, therefore, that iron sometimes flies and sometimes follows the magnet, was scarcely justified by his experimental facts, and it is a mistake to suppose, as some have done, that he was aware of the polarity of permanent magnets.

If we tap the card in the last experiment a curious result may sometimes be observed. The lines of filings will be seen to recede from the point of the card immediately over the pole of the magnet. If, however, the magnet be held over, instead of under, the card, tapping will cause the filings to approach the point under the pole of the magnet. The most probable explanation of this is to be found in the fact that the erected filings stand in the former case as shown in fig. 18, and in the latter as shown in fig. 19; that is, in both cases, owing to the action of gravity, they are more acutely inclined to the card than the lines of force (represented by dotted lines in the figure). Consequently, when the filing springs up into the air, and is thus free to follow the magnetic couple, it turns more into the direction of the line of force; the effect of this is to carry its lower end each time a little farther from the axis of the magnet in the one case, and a little nearer to it in the other.

By far the most important case of magnetic induction is the electromagnet. Whenever an electric current flows in a closed circuit, the surrounding space becomes a field of magnetic force, and any piece of iron in it will be inductively magnetized. Such an arrangement of an electric circuit and iron is called an electromagnet. The variety of form and of application of such instruments in modern science is endless. A few of the more important modifications will be considered below.

Co-existence of Induced and Permanent Magnetism.— The fact that a body is already a permanent magnet does not prevent its being susceptible to magnetic induction. If we take any piece of iron at random, the chances are that one end or other of it will repel the north pole of a magnetic needle,—in other words, it will be to some extent permanently magnetic; but if we bring it slowly nearer and nearer to the pole of the needle, provided its magnetism be not too strong, it will by and by attract the pole which it at first repelled. Again, if we take two steel magnets, which may be as powerful as we please, provided at all events that they are unequally powerful, and bring two like poles together, these poles will at first repel each other in accordance with the fundamental law of permanent magnets; but, when the distance is less than a certain amount, the repulsion passes into an attraction, and when the poles are in contact this attraction may be very considerable. These phenomena are at once explained by the law of induction. The induced or temporary magnetism is superposed on the permanent magnetism, and, when the poles are near enough, the opposite magnetism induced by the pole attracts it more than the permanent like magnetism repels it; and this happens even with steel, whose susceptibility for magnetic induction is considerably less than that of iron. This phenomenon was observed pretty early in the history of magnetism, but was not fully explained until the idea of magnetic induction was fully developed. Michell, in his Treatise of Artificial Magnets, gives a tolerably clear account of it. Musschenbroek mentions it, along with the fact that a magnet attracts iron more than it does another magnet, but offers no explanation of either fact. The latter result, so far as it is true, can of course be explained by the smaller susceptibility of steel, particularly of hard steel, to magnetic induction, which is the main factor in attraction at small distances. Poggendorff and others have experimented on the subject in later times. The reader should notice the close analogy between these phenomena and the repulsion and attraction at different distances between two similarly electrified conductors. See article Electricity, vol. viii. p. 33.

Induction of Permanent Magnetism.— The case above supposed, in which the induced magnetism is wholly temporary, although it can be easily realized with small magnetizing forces, is not the general one, but in fact the exception. Usually a certain proportion of the magnetism remains after the inducing farce is removed. This happens even with the softest iron, when the inducing force is very great. Just as bodies differ very much in their susceptibility for induced magnetism, so they differ greatly in their power of retaining this magnetism when the inducing force ceases, or, as the phrase is, in “coercive force.” Thus, while the inductive susceptibility of steel is less than that of iron, it retains much more of the magnetism imparted to it, and is therefore said to have much greater coercive force; and the coercive force is greater the harder the steel is tempered.

It is obvious, therefore, that the principle of “induction,” along with the idea of “retaining power” or “coerciveforce,” furnishes us with the key to the explanation of the communication of permanent magnetism, whether by means of natural magnets or of artificial magnets, or of the electric current. In particular, we see at once the reason why the end of a needle which has been touched by the north pole of another magnet becomes a south pole, and vice versa,— a fact which greatly puzzled the earlier magnetic experimenters, and indeed all who were inclined to think that, in the process of magnetization, something was communicated from the one magnet to the other.

Mathematical Theory of the Action of Permanently Magnetized Bodies.

In this section we shall suppose the bodies considered to be rigidly magnetized; i.e., we shall suppose that magnetic action exerted on any body produces no change in its magnetization. It is further to be observed that we are merely establishing a compendious representation of observed facts, and foreclosing nothing as to their physical theory or ultimate cause. Our method is therefore to some extent tentative, and its success is to be judged by the agreement of the results with experiment.

There are two main facts to be borne in mind :—(1) that a magnet is polarized, and (2) that the properties of its smallest parts are similar to those of the whole. Adopting the mathematical fiction of action at a distance, we may represent the action of such a body by a proper distribution of imaginary positive and negative attracting matter throughout its mass. This imaginary matter, following Sir W. Thomson, we shall call “magnetism,” as we thus avoid suggesting other properties of matter than attraction, of which in the present case experience has given no evidence. We assume that magnetism of any sign repels magnetism of the same sign and attracts magnetism of the opposite sign. Magnetism is supposed to be so associated with the matter of the body that magnetic force exerted on the magnetism is ponderomotive force exerted on the matter. On the other hand, magnetic force is always supposed to be exerted by magnetism upon magnetism, and never directly by or upon matter. Into the nature of this association of magnetism with matter there is no pretence, indeed no need, to enter.

The elementary law of action assumed is that the attraction or repulsion (as the case may be) between two quantities m and m' of magnetism supposed concentrated in two points at α distance r apart is and is in the line joining the two points. This supposes that the unit quantity of magnetism is so chosen that two units of positive magnetism at unit distance apart repel each other with unit force. This definition, which is fundamental in the electromagnetic system of units, gives for the dimensions of a quantity of magnetism [lJM i T -1 ]. If the electrostatic system be adopted the result would of course be different.

An accurate meaning can now be given to the phrase “strength of a magnetic field,” or its equivalent “resultant magnetic force at a point in the field;” it is defined to be the force exerted upon a unit of positive magnetism supposed concentrated at the point. The force exerted on a unit of negative magnetism would of course be equal in magnitude, but oppositely directed; and in general, if R denote the resultant magnetic force at the point, the magnetic force exerted on a quantity κ of magnetism concentrated there is κR,

We may, as in the corresponding theory of electricity, introduce the ideas of volume density (p) and surface density (σ),—so that ρdν and σdS denote the quantities of magnetism in an element of volume and on an element of surface respectively; p and σ may of course be positive or negative according to circumstances.

It will now be seen that, mathematically speaking, the theories of action at a distance for electricity and magnetism are identical, and every conclusion drawn will have, so far as the physical diversity of the two cases may allow, a double application. In particular it will be found that the theory of magnetism, when properly interpreted, gives the theory of dielectrics polarized in the way imagined by Faraday.

The fact of magnetic polarity requires the conception of negative as well as positive magnetism; the fact that the properties of the smallest parts of a magnet are similar to those of the whole requires that in every element of the body there shall be both negative and positive magnetism. From the fact that in a. uniform field, i.e., one in which the resultant magnetic force has at every point the same magnitude and direction, the force of translation upon a magnet is nil, it follows that the algebraic sum of all the magnetism in any magnet must be zero; for, if R denote the strength of the field, by the theory of parallel forces the whole force on the magnet will be S(κR), = R≤κj hence ‰ = 0. In other words, in every magnet there must be as much negative as positive magnetism ; and this conclusion also must be extended to the smallest parts of every magnet, so long as we do not go behind the mere facts of observation. The positive and negative magnetism cannot be coincident throughout, otherwise there would be no external magnetic action, but the separation is in the elements of the body. Thus, although there is no force of translation in a uniform field, there will in general be a couple. Consider the positive and negative magnetism separately, and let κ denote any element of the former and κ any element of the latter. Let N be the centre of mass of the positive, S the centre of mass of the negative magnetism; so that, if the magnet be referred to a set of rectangular axes, the coordinates of N and S are 2κχ 'S.κy ½κz

2/c ’ 2/c ’ 2,κ

and (1).

2∕c⅛' 2∕c'¾∕ 2∕c'√

2/c' ’ 2/c' ’ 2κ' -

Let the distance NS = l, and let K = Γ½κ, = - ι⅜κ,' ; this quantity K is called the “magnetic moment.” By the theory of parallel forces, if we suspend the magnet in a uniform field of strength R, the action upon it reduces to two forces R≤κ and - R2∕c, each parallel to the direction of the field, acting respectively at N and S, in other words to a couple whose moment is RSκZsinχ or KRsinχ, where χ is the angle between SN and the direction of the field. Hence, if the magnet be perfectly free to follow the magnetic action >→ ' >—► of the field, it will set so that the linβ SN or the line NS is parallel to the direction of the field, the equilibrium being stable in the former case, but unstable in the latter. The line NS is therefore parallel to what we have already defined on experimental grounds as the axial direction in the magnet. N, S, and NS are sometimes called par excellence the poles and the axis of the magnet; we have adopted the looser definition given above because it is more convenient and nearer the popular usage.

The above results maybe applied to some cases very important in practice. Let the magnet whose centres of positive and negative magnetism are N and S be suspended by the middle point of NS, which, for simplicity, may be assumed to be also its centre of gravity. Let OX (fig. 20) be a horizontal line drawn northwards, OZ

a vertical drawn downwards, both in the magnetic meridian. Let the vertical plane through NS make an angle 0 with the magnetic meridian, and let ON make an angle <ρ with the horizon. If R be the strength of the earth’s magnetic field, and ι the angle of dip, then the horizontal and vertical components of the eartn s force are H = R cos» and Z = R sinι.

First, suppose the angle φ fixed, and the magnet free to rotate about OZ only; then the couple tending to diminish the angle 0 is 2.2/c. R cost-^- cosφ sinø, or KR cost cos≠ sinø.

In other words the directive couple varies as the sine of the angle of deviation from the magnetic meridian. This conclusion was verified experimentally by Lambert, and also by Coulomb by means of his torsion balancé. It will be seen that, eaeteris paribus, the directive couple is greatest when the magnetic axis is horizontal.

Next suppose the angle θ fixed, and the magnet free to rotate about a horizontal axis inclined at an angle 90°- 0 to OX. The couple tending to diminish the angle φ is KR (cost cos θ sin φ - sin i cos<∕∣). The position of equilibrium is given by the equation tan φ=sec Øtant.

The angle at which the axis is depressed below the horizon is therefore least when 0 = 0, and greatest when θ = 90° j its value being ι in the former case, and 90° in the latter, as stated above, p. 221.

In general, if λ', μ', v ’ and λ, μ, v be the direction cosines of the direction of the field and of the axis of the magnet respectively, then, resolving the forces acting at N and S, we see at once that the three components of the magnetic couple are

KR(∕∕x- μ'v), KR(λ ’ p — ∕λ), KR(μ'λ-λ⅛) . . . (2).

These are clearly the same as the components of the couple on a system of three magnets whose axes are parallel to OX, OY, OZ, and whose magnetic moments are Kλ, K∕z, Kv. Hence, so far as the action of a uniform field is concerned, we may resolve the magnetic moment like a vector, and replace a given magnet by others the resultant of whose moments is the moment of the given magnet.

It appears therefore that in a uniform field every magnet behaves as if it were made up of a certain quantity of positive magnetism and an equal quantity of negative magnetism placed at such a distance apart on a line parallel to the magnetic axis that the product of the quantity of magnetism into that distance has a value equal to the magnetic moment of the magnet. It is very important to observe that the magnetic moment alone appears in the above formulae for the magnetic action. We cannot therefore separately determine from observations in a uniform field either the quantity of positive or negative magnetism in a magnet or the distance between the magnetic centres of mass.

Let M be any magnet, and P a point whose distance from any point of M is infinitely great compared with the linear dimensions of M. Then, since all the lines drawn from P to different points of M are sensibly parallel and equal in length, we may suppose the positive and negative magnetism of M to be collected at their mass centres, i.e., M to be replaced by an ideal magnet. It is also obvious that, throughout a region around P whose linear dimensions are of the same order as those of M, the field due to M may be regarded as uniform. Hence we conclude that in calculating the mutual action of two magnets M and M' we may replace each of them by an ideal magnet, provided the distance between them be infinitely great compared with the linear dimensions of either. This condition may be satisfied either by making the distance between the magnets very great if their dimensions be finite, or by making their dimensions infinitely small if the distance between them be finite. The second alternative suggests at once a method for representing the magnetic action of magnetized bodies at finite distances. We may divide up the body into portions whose linear dimensions are infinitely small compared with their distance from any point at which their action is to be considered; each of these portions is itself a magnet, and may be replaced by an ideal magnet having the same axis and moment. The whole magnetic action is obtained by integrating the action of all the ideal magnets of which the body is thus supposed to be composed.

Let λ, μ, v be the direction cosines of the magnetic axis of any element dv of a magnet, and I such that Idv is the magnetic moment of the element, and let Iλ = A, Ip. = B, Iv = C; then I is called the “intensity of magnetization” at the point where the element is taken. I may be regarded as a vector which specifies the magnetization of the body; in general it varies continuously from point to point; if it has the same value and direction at every point, the body is said to be uniformly magnetized. A line drawn so that the direction of I at every point of it is tangential to it is called a “line of magnetization.” It is clear from what has already been shown that we may if we choose replace the element dv by three ideal magnets whose axes are parallel to the coordinate axes, and whose moments are A.dv, B<Zv, Cdυ respectively.

If then K he the magnetic moment of the whole magnet, δK the moment of any element δυ, and p, q, r the direction cosines of the axis of the whole magnet, we have K=Z2κ, = - Z2κ'; and, remembering that κ'= — κ for every element,

-A ¾-¾ f ∖

_(2kx 2k⅛'∖1 - 5 ( k ∙ i 7_2(δΚλ)_2ΐλδν 2Aδv P ∖Sκ 2√ J l~ l⅛κ K - K “K '

We may therefore write, replacing summation by integration, Vp=fffkdv, lkq=fffV>dv Kq=JJfCdv . . (3).

Let SN be an ideal magnet of infinitely small length l, let m be its-magnetic moment, and rn = κl. Let Q be its middle point, and the angle PQN = θ, N being the positive or north-seeking pole; and let QP = D. Then the potential at P due to this magnet is κ{ D 2 -DZcosβ + ∣Z 2 } - *-κ{ D 2 + DZcosβ + ∣Z 2 }^ i .

Expanding and neglecting powers of above the first, we get for the potential mcosθ ∕.x

D 2 λ

Hence the potential at P (ξ, η, f) of an infinitely small magnet Adv at (x, y, z), having its axis parallel to the axis of x, is A (ξ-ie)∕D 3 , and similarly for the other two. We therefore obtain for the potential of the whole magnet

{A(ξ -«) + B(η - y) + C(C - ≡)

-JΓf ∖ a ⅛) +b ⅛) + 0 ⅛) I , ir > < i >∙

JJJ ( dx dy dz )

∕77* τ (d d ) 1 ,

=∕∕∕ I j λy + μ-j — h ν- r ( ~τ^dν J

JjJ ( dx dy dz ) D j

Taking the second of these expressions and integrating by parts in the usual way, we get y, -f∕i is ∙Jff^' where σ = ZA + wιB + nC = Icos0 . . · . (6),

= ∕⅛A,≤B rfC∖ p ∖ote *dy + dz) J l, m, n being the direction cosines of the outward normal to any element iZS of the surface of the magnet, and θ the angle between the normal and the direction of magnetization at <ZS.

Hence the action of any magnet may be represented by means of a certain volume distribution (p) and a certain surface distribution (σ) of free magnetism. This important proposition is due to Poisson.

The fact, in itself obvious, that the sum of all the magnetism of Poisson’s distribution must be zero, gives the theorem

>^(⅛ + ≡ + ^>^ a+,,,b+, * c >" s ) (;)i =^IcosfldS )

which admits of course of direct analytical proof.

The magnet may also be replaced, so far as its external action is concerned, by a distribution wholly on its surface, as was shown by Gauss. This will be seen at once if we replace the positive and negative magnetism throughout the body by positive and negative electricity, and suppose the surface of the magnet covered with a conducting layer in connexion with the earth. The surface will thus become charged with a distribution of positive and negative electricity whose total sum is zero, such that the potential of the surface is zero, and hence the potential at every external point zero. The potential of this surface layer at every point external to the body is therefore equal and opposite to that of the internal electricity. If, therefore, we change the sign of the surface density at every point, we obtain a surface distribution whose potential at every external point is the same as that of the body. There is of course only one such distribution: we may call it Gauss’s distribution.

Poisson’s distribution will coincide with that of Gauss provided the magnetization be such that

<ZA <ZB dC _ ∕RV

dx dy dz when this condition is satisfied at every point of the body, A is said to be “solenoidally” magnetized; a particular case is that of uniform magnetization.

So long as the point considered is external to the magnet there is no difficulty in attaching a definite meaning to the resultant magnetic force (∣∣) at a point; its components are given by dV n dV dV , ov α =-jΞ> 3--7F-» 7=--j- · · · (9 ); dx dy ’ dz and the values obtained will be the same whether V be calculated by means of Poisson’s or of Gauss’s distribution. Inside the body the result is otherwise, for reasons that are not difficult to understand, when we examine the nature of our fundamental assumptions. It is therefore necessary to be careful to define what we mean by resultant magnetic force in the interior of a magnet. It is defined by the above equation (9) on the understanding that V is calculated from Poisson’s distribution. We can show that ∣j thus defined is the resultant force in an infinitely small cylindrical cavity within the magnet, whose axis is parallel to the line of magnetization, and whose radius a is infinitely small compared with its axis 2δ.

The removal of the matter filling such a cavity will affect Poisson’s volume distribution to an infinitely small extent; the alteration of the force if any, will therefore arise simply from the surface distribution which we must place on the walls of the cavity in order to make up the complete representation of the action of the magnet in the cavity. This distribution reduces to two circular disks of radius α at the two ends, the densities of the magnetism on which are - I and +1 respectively. The action due to these is a force 4πl(l - δ∕Vα 2 + δ 2 ) in the direction of magnetization. If a be infinitely small compared with b, this force becomes zero, which proves our proposition.

If, on the other hand, the cavity in the magnet be diskshaped—say a narrow crevasse perpendicular to the line of magnetization—then the force due to the distribution on its walls becomes 4ττl, and the resultant force in the cavity is no longer ∣j, but a force g, whose components are α=α+4πA, J=β + 4πB, c = *y + 4τrC . . (10).

is called the “magnetic induction” at the point (x, y, s). From the definition of ⅜) it follows that outside the magnet

4± + 4⅛ + ⅛. 0 (U).

dx dy dz

Inside ⅛

dα dβ d y dx + dy + dz≈ πp )

> · · (12).

j fdK dB rfC∖(

= ^ 4τ U¼⅛)

At the surface of a magnetized body the tangential component of ∣j is continuous, but the normal component increases abruptly by 4τrlcos0 in passing from the inside to the outside of the surface.

Outside magnetized matter the magnetic force and the magnetic induction are coincident. Inside we have da db , de da dβ dy . ∕dA , dB dC∖ n , 1 < n

3—+ 5— + 3~ γε = ,- ; h-,—h 4τr I ——f-- F-r^ I - "· (*·“)·

dx dy dz dx dy dz ∖ dx dy dz ∕

Hence the magnetic induction satisfies the solenoidal condition both inside and outside magnetized matter. It has normal continuity, and, in general, tangential discontinuity, at the surface of a magnetized body.

For if V, τ and n, t be the normal and tangential components of and $ just inside, and √, √ andτι', t' the corresponding components j ust outside the surface near any point, we have n = v + 4∙∏-I cos β, and n'=v' ∖ but √=v + 47rlcos0, therefore n=n'. On the other hand √=Z ’, whereas τ is the resultant of t and 4πlsin0, which is parallel to the surface, but otherwise may have any direction according to circumstances; hence, since t = t, in general τ' is not equal to τ.

In fact there will be tangential discontinuity of the magnetic induction unless the line of magnetization be perpendicular to the surface of the magnet; in this case there is complete continuity of the magnetic induction. When the magnetization at the surface is tangential, there is, on the other hand, complete continuity of the magnetic force.

It follows from the above that the surface integral of the magnetic induction taken ov⅛r any closed surface S vanishes.

First, let the surface be wholly within or wholly without continuously magnetized matter. We have, integrating all over S and all over the space enclosed by S, the analytical theorem

^ fa ÷" ii +^ s ^(S + S + ⅛>∙ · (14)! hence the result follows, for every element of the right-hand integral vanishes. Next, suppose S to be partly within and partly without a magnetized body. Divide it into two parts by a double partition one of whose walls runs outside the surface of the body and infinitely near it, the other inside and infinitely near it; then, on account of the normal continuity of the surface integral will be the same in absolute value over each of these walls. Hence the integral over the whole of S differs infinitely little from the sum of the integrals over the two surfaces into which it is broken up by the double partition, each of which vanishes by the former case. Hence the theorem holds in this case also.

We may therefore apply to lines and tubes of magnetic induction without restriction all the theorems proved for lines and tubes of electric force in space free from electrified bodies. We may speak of the number of lines of magnetic induction instead of the surface integral if we choose. And we have this important theorem :—

The number of lines of magnetic induction that pass through an unclosed surface depends merely on its boundary.

There must therefore be a vector JA, whose line integral round the boundary is equal to the surface integral of $ over the surface.

The components F, G, H of §1 are connected with those of $ by the equations = dH_dG Ä= dF_dH = rfG_dF fl ∩ α dy dz ’ dz dx , C dx dy ° ’ as has been shown in the article Electricity, vol. viii. p. 69.

Mutual potential energy and mutual action of two magnetic systems.— The potential energy of a small magnet is κ(V 2 -V 1 ), where V 1 and V 2 are the values of V at its negative and positive poles. If the magnet be infinitely small, of length ds say, the direction cosines of ds being λ, μ, v, this may be written κdsdT∕ds, i. e., mdV∕ds, or, if we are considering a magnetized element of volume dv, τ ∕ dV 1 dV 1 dV ∖ , πβλ

I ( λ-τ- + μ-τ- + y- τ - ∖ dv (16).

∖ dx r dy dz J

Hence the potential energy of the whole magnetic system in a field whose potential is given by V is

-≡÷≡l÷⅜) πn

∕7Z ( · · · (1/)>

= - Zy(Aα + Bj3 + C7)<Zr )

the integration being extended all over the magnetized masses supposed to be acted upon. Integrating by parts, we get at once W=ffVσdS +JP^Ngdv (18),

<r and ρ being the surface and volume densities of Poisson’s distribution, a result that might have been expected. W may also be expressed as a sextuple integral; for, if Γ, λ', f, ι', x', if, z ’ refer to the acting system, then x ~ff' 1 '(λ 'έ + '∙⅛ + ''<⅛H*' i ' j,w

Whence w (λ ⅛ + * t ⅛ + )

? (iy).

( , d ,d , d ∖ 1 I

x ∖ λ dx' + μ df^ h ^dz'jD ' A remarkable expression for W may be obtained by supposing the integration in (17) extended throughout the whole of space, on the understanding that A, B, C are zero where there is no magnetized matter, and then integrating by parts. We get, since the surface integral at infinity may be shown to vanish, pol

00 ( · · (- ’ θ)> ..m- v (^Adc ∖ ιlv JJJ ∖ dx cly dz J — <X) where it must be understood that A, B, C vary continuously, however rapidly. In point of fact, where, as at the surface of a magnetized body, there is discontinuity, a finite portion of the integral will arise from an infinitely thin stratum near the surface. The proper representation of this part will be a surface integral, as may be seen by referring to (18), from which we might have started.

If now V ’ be the potential of the magnet acted upon, then d 2 V ’ <Z 2 V' dfVj (dh. cZB dC∖ . dx i + dy i + dz i “ ∖ dx + dy + dz J ’ whence 00 w 1 ΓΓΓxr( d ^' , rf2γ ' , ^ 2γ, ∖ . ι

w ^^JiZz7 v (di r+ ⅛ 5 ' + * r ∕ fo

, 1 /777<* γ dV' dV dV ’ , dy dV' ∖ , + ^JjJ ∖dx dx + dy dy dz dz.) ' ( 21 ) ’ — 00

00

= + ~ cosø dv -∞ where R and R' are the resultant forces at any point of space due to the acting and acted-upon systems respectively, and θ the angle between their directions.

In practice W is expressed as a function of the variables (equal in number to the degrees of freedom) that determine the relative position of the two systems; differentiation with respect to any one of these then gives the generalized force component tending to decrease that variable.

We may also calculate the forces directly. For, the components of force on the element dv, being the differences of the forces acting on the two poles of the element, are ( .da , τ ,dα r .da.\. .

I A- r -+ B- y - + C- y - ]dv, &c.; ∖ dx dy dz J ’ and the components of couple, in calculating which the field may be supposed uniform, are (see above, p. 228)

(γB-j3C)<⅛, &c.

Hence, integrating, we get, with the chosen origin, for the components of the whole force and couple, and similarly for g and g.

s-jΨb- 3 °+ 1 ,( a ⅛ + b ⅛ +c ⅛) · < 22 >∙ and similarly for and $.

In the important case of a uniform field whose components are α, β, γ, we have

W= - ^K(la + mβ + nf} (23),

K being the moment of the magnet, and l, m, n the direction cosines of its axis. From this formula the results given above (p. 227) can be deduced with great ease.

Examples.— Some examples of the application of the foregoing theory are here given, partly on account of their intrinsic value as types enabling us to conceive the different varieties of magnetic action, partly for the sake of the light they throw on the theory itself. The reader who desires more such should consult Maxwell’s Electricity and Magnetism, or Mascart and Joubert, Leçons sur V Électricité et le Magnétisme. Solenoidal Magnets have already been defined as such that the vector I satisfies the solenoidal condition dA dB dC - r + j- + ^- = 0 ∙ dx dy dz

The lines of magnetization, therefore, have all the properties of lines of magnetic induction or electric force. In particular, if we consider a portion of the magnet enclosed by a tube of the lines of magnetization, the product of the intensity of magnetization by the section at each point is the same. Such a portion of magnetized matter taken by itself is called a “magnetic solenoid,” and the product mentioned is called its “strength.” It is clear (from the general definition, or it may be proved directly from the secondary property just mentioned) that the action of the solenoid may be represented by the distribution of a certain quantity ωl of positive magnetism on the one end and an equal quantity of negative magnetism on the other, I being the intensity of magnetization, ω the normal section at the end. The action therefore depends merely on the strength of the solenoid and on the position of its ends. The shape of the intervening portion is immaterial. If we suppose it straight, and if the section be infinitely small so that the magnetism at the ends may be regarded as condensed at two points, we have an ideal magnet of finite length. The equipotential lines of such a magnet in any plane through its axis are of course given by the equation - ^^r = const (24), where r and r' are the distances of any point P on the line from the poles.

The equation to the lines of force is easily obtained; for, if NP and SP (fig. 21) make angles θ and 0' with the axis of the magnet, and φ and φ' with the line of force, we must have sin≠∕j∙ 2 - sin φ ’ ∣ r 2 = 0; hence, since s, ∖ Αφ=rdθ∕ds , sin≠' = r f dtf∕ds, we get dθ ∣ r - dθ' ∣ r ’ °≡∙ 0; i. e. , sin θdθ - sin θ ’ dθ ’ = 0; which gives for the equation to a line of force cos 0 - cos 0'—const (25).

We may imagine a magnet of this kind so long that the action of one of its poles may be altogether neglected at points which are at a finite distance from the other. We thus effectively realize what never occurs in nature, viz., a magnet with one pole only. If we place the like poles of two such magnets near each other, we get a field the equipotential lines and lines of force in any axial plane of which are given by the equations

JL + JL = const (26).

r τ cos0+ cos 0 ’ = const (27).

The lines of force given by equations (25) and (27) may be traced in a diagram by means of the following simple and elegant construction Draw two circles A and B, having equal radii and N and S respectively for centres; produce the line NS both ways, and, starting from the centre, divide it into any number of equal parts; through these draw perpendiculars to meet the circles A and B; from N draw a series of lines to the points of division on B, and from S a similar series to the points of division on A. These lines will form a network of lozenges the loci of the vertices of which will be lines of force, corresponding to (25) or (27) according as we pass from point to point along one set of lozenge diagonals or along the other. Fig. 22 will give the reader an idea of the general appearance of the two sets of lines. He may compare the ideal with the actual cases by referring to figs. 4 and 5, p. 222.

I ιι the case of an infinitely small magnet, the equipotential lines are of course given by the polar equation r 2 =c 2 cos θ, c being a variable parameter. It is easily shown that the lines of force, which are necessarily orthogonal to these, have for their equation r=c sin 2 0. If φ be the angle between r and the tangent of the line of force, we have ta,nφ=rdθ∕dr≈⅛ tanø; hence the following construction for the direction of the line of force at P due to a small magnet at 0 :— let K be the point of trisection of OP nearest 0, and let KT, per-pc ιdicular to OP, cut the axis of the magnet in T; then TP is the tangent to the line of force at P. This construction in a slightly diiferent form was given by Hansteen and by Gauss; the latter ad Is that the resultant force at P is given by M.PT∕0T.0P 3 where M is the magnetic moment of the magnet, a proposition which the reader will easily verify. These propositions are of considerable use in rough magnetic calculations. As this is an important case we give a diagram of the equipotential lines and lines of force in fig. 23.

We may, if we choose, consider a filament of matter magnetized longitudinally at every point, but so that the strength ωl ( = J, say) is variable. Such a filament is called a complex solenoid. It may clearly be supposed made up of a bundle of simple solenoids whose ends are not all coincident with the ends of the filament. If ds be an element of such a filament, the potential is given by τr ∕" τ ^∖T)∕ J 1 J 2 Λl dJ , osλ y f d,3 -ΛΓ-D l -DH ds D Z < 28 >'

That is, its action may be represented by two particles of magnetism J 1 and J a at its two ends, and by a continuous distribution <>f free magnetism along its length whose density is -dJjds. This is of course merely a particular case of Poisson’s distribution.

When a body is solenoidally magnetized, the magnetic force H both external and internal depends solely on the surface distribution, i.e., merely on the ends of the solenoids of which the body is composed. We may therefore suppose the two ends of any solenoid joined by a solenoid of equal strength lying in the surface of the body. Proceeding thus, we may in an infinite number of ways construct a surface layer of tangentially magnetized matter which will represent the magnetic action of a solenoidally magnetized body. Thomson has shown by means of a highly interesting piece of analysis how to find the components of this tangential magnetization. See Reprint of Papers on Electricity and Magnetism, p. 401.

The magnetic theorems just stated will suggest at once to the mind of the reader acquainted with the analysis employed in hydro kinetical problems the close analogy that subsists between the two methods. In fact, by proper arrangement, every problem in the one subject can be converted into a problem in the other. For details we refer the reader to Thomson, who was, so far as we know,;he first to work out this matter fully; in the present connexion lie should consult more particularly §§ 573 sq. of the Reprint. Uniformly Magnetized Bodies constitute in practice the mo>t important ease of solenoidal magnets. In the first place it is obvious that the whole magnetic moment ρf such a body is simply its volume multiplied by the intensity of magnetization, and that the axis of the whole is parallel to the axis of each of its infinitely small parts. The method usually applied to calculate the potential in this case may be presented in two ways. The potential is calculated according to Poisson’s method in this case merely from a surface distribution of varying density I cos θ. We may replace this by a layer of uniform density ρ and varying normal thickness. Let the thickness at any point measured parallel to the magnetic axis be t; then the normal thickness is icosø; hence ptcosθ≈ Icosθ, and ρt≈l; i.e., t is constant. We may therefore suppose the magnet replaced by itself (fig. 24) with a uniform volume distribution p of positive magnetism, and itself displaced through a distance t in a direction opposite to that of magnetization with a uniform volume distribution - p ; or, which comes to the same thing,' the potential of the magnet at P is p(U - IF), where U is the potential at P of a uniform volume distribution of density +1 throughout the magnet, and IF the potential of the same at a point P' displaced through a distance t in the direction of magnetization.

If l, m l n be the direction cosines of the magnetic axis, this gives at once „ fdu fj , dir ,du, ∖)

v --4⅛ h+ ⅛'" + ≡ m )

__ 1 (Ae + „^ + ,Ae) · · · < 28 >> ∖ dx dy dz j

= AX + BY + CZ

where X,Y,Z are the components of the resultant force due to volume distribution p = +1 throughout the body, and A, B, C the components of the magnetization.

The same result may also be arrived at thus. The part of the potential due to the element dv is Idvcos θ∕r 3 , but this is the component parallel to the direction of I of the resultant force at P of a volume distribution whose density in dv is I; hence, since the direction of I is everywhere the same, the whole potential is the component parallel to the magnetic axis of the body of the resultant force at P of a volume distribution p = I throughout its whole extent. This gives at once the expression of (29) for V.

In the case of a uniformly magnetized sphere of radius a, the axis being parallel to the axis of x, and the centre at the origin, we get, if r be the distance of P from the origin, for external points,

V≈⅜ιrlα i ⅛∕r 3 (30); in other words, the external action is that of a magnet of infinitely small dimensions, having the same moment and axis, placed at the centre.

For internal points

V=∣<Iaj (31); whence it appears that the magnetic force inside the sphere is constant in magnitude and in direction, being opposite to the uniform magnetization, and equal to -⅜τrl.

The potential of a uniformly magnetized ellipsoid may be similarly treated. Let the origin be at the centre, and the axes along the principal diameters of the ellipsoid, whose lengths are 2α, 2Z>, 2c, and let l, m, n be the direction cosines of its magnetic axis. Consider first an external point. Then, if

T— d ≠ M=&c., N = &c. (32),

Λ √(α 2 + 4>)∙' ’ (⅛ 2 + ≠)(c 2 + 4>)

where α is the positive root of ae 2 y 2 z 2 __

a 2 + φ b' 2 + φ + c 2 + Φ * we have

X = Lat, Y=My, Z = Nz, and

V=ALat + BMy + CNz (33), where it must be remembered that I., M, N are functions of x, y, z, inasmuch as a is so.

If (x, y, z) be an internal point, X, Y, Z are the components of the force due to a similar and similarly situated ellipsoid through (x, y, z). Let its axes be pa, pb, pc ; we now have

X = Lx, Y = My, Z = Nz, where

I. = 2τrpafbpcf .

V o ∖ l(p 2 a 2 + φ) 3 (p 2 b 2 + φ)(p 9 c 2 + φ) or, writing φ =p 2 ψ, I. = 2πabc∕‰==Λ==, M=&c., N=&c. . (34).

Λ √(α 2 + ψ) 3 (6 2 +ψ)(c 2 + ψ)

We thus obtain for V,

V = ALa: + BMτ∕ + CNz, (35), where I., M, N are now constants, which remain the same so long as the ratios of the axes remain unaltered. The components of the force inside the ellipsoid are

α=-AL, j8=-BM, 7=-CN . . . (36).

The force is therefore uniform; but its direction does not coincide with that of the magnetization, unless the latter be parallel to one of the principal diameters, and then the force is opposite in direction to the magnetization. It will be observed that the force inside similar ellipsoids similarly magnetized to the same intensity is always the same.

For an oblate ellipsoid of revolution, in which b=c=a[*fle 2 , τ ,∕ι √Γ≡P. 1 ∖ „ xr n ∕√Γ^7 2 . 1 ι-c 2 ∖ l = 4 ∙∖^^ e 3-sm- 1 e), M = N-2tr Ç sιn^ 1 ej. For a very flat oblate ellipsoid of revolution I. = 4τr, M = N = ιr 2 α∕r. For a prolate or ovary ellipsoid of revolution, in which a = b = c V1 - e 2 ,

N~u(-1-1 )(∣⅛⅛-1)∙

From the formnlae for an ellipsoid we could easily deduce those for an infinitely long elliptic or circular cylinder; we have merely to make one of the axes infinite. We find in this way, for instance, that the force inside a circular cylinder of infinite length magnetized transversely is - 2πl.

The reader will find it interesting to examine the values of the magnetic induction in the foregoing cases, and to verify its normal continuity at the surface of the magnet.

Lamellar Magnets form another very important class. In them the components of magnetization are derivable by differentiation from a function φ(x, y, g), which is sometimes called the “potential of magnetization,” so that

Λ-⅛, B-⅛, C-⅛ (37).

dx dy dz

It is obvious at once that the family of surfaces φlx, y, z) = const, cut the lines of magnetization at right angles; for, if dx, dy, dz be the projections of the element of any line on the surface, we have by differentiation dφ , dφ , dφ , f-dx+-f-dy + -jdz=O , dx dy dz ’ i.c., Adx + Tidy + Cdz≈0 , which is the analytical expression of the property in question. We may therefore suppose a lamellar magnet divided up by these surfaces of magnetization into an infinite number of infinitely thin normally magnetized shells or lamellae. It can be shown that the product of the intensity of magnetization by the thickness at each point of any such shell is the same; for, if φ(x, y, z) = c and φ(x, y, z) = c + Sc be the equations to the two surfaces bounding the shell, δv the normal distance between them at any point, we have

⅛x + ¾ i , + ¾ 2 -Sc, dx dy dz hence ( dφ dx dφdy dφ dz ∖ dx dv dy dv dz dv) v c ’ i.e., Iδv=δc=constant for the same shell, which was to be proved. This product is called the strength of the shell.

A shell, which is everywhere normally magnetized, but whose strength is not constant, is called a ‘ ’ complex shell”; a magnet made up of such shells is called a “complex lamellar magnet.” The condition to be satisfied by A, B, C in this case is simply that the lines of magnetization must be orthogonal to a family of surfaces, t.β., Aifø + Btty + Cifz must be convertible into a perfect differential by multiplication by a factor; otherwise that

./dB dC ∖ τι fdC dk ∖ HdA tZB∖ λ a (≡-⅛) +b (⅛-≡) +c (⅛-^)- 0 ∙

The potential at P of a simple magnetic shell of strength i is given by the formula

, r ∕Z icosβ ^ s · ∕7 r dScos0

n ~J∕ ^τ> r ^ ~fj · < 38 >· where ω is the solid angle subtended at the point P. There is a convention here as to sign, viz., that side of the shell is positive towards which the lines of magnetization pass, and the solid angle subtended at points infinitely near that side is positive, while that subtended at points infinitely near the other side is negative. If we cause P to move from the positive side away to infinity, then back from infinity to the negative side, or to move anyhow from infinitely near the positive side to a point infinitely near the negative side without cutting through the shell, it will decrease continuously by 4πi; if we pass through the shell from a point infinitely near on the negative side to a point infinitely near on the positive side, there will be a sudden increase of 4πi ; tangentially to the shell there is continuity. The potential of a closed shell is evidently zero for any external point, ±4τrt for an internal point according as the positive or negative side is innermost. It appears also that the potential of a simple magnetic shell depends merely on its strength and on its boundary, just as that of a magnetic solenoid depends merely on its strength and the position of its ends.

A lamellar magnet will in general ’ be made up partly of closed shells, and partly of shells wlιose boundaries lie on the surface; only the latter of course can influence the potential at external points. The general expression for the potential at any point T(ξ, n, 0 is v.zy∕⅛ i⅛) + ⅛ li⅛) + ⅛ d (⅛λ fι ,.

jJJ ∖ dx dx dy dy + dz dz J where θ is the angle between D and the outward normal to dS, and φ ’ the value of φ at the point ξ, η, ζ (zero of course if ξ, η, ζ be outside the magnet). The value of V thus found is not discontinuous at the surface as might be supposed, for both the surface integral and 4∏∙φ' have discontinuities there, and they are equal in amount and of opposite sign.

For an external point the potential is

The immediate interpretation of this is that the potential is the same as that due to a normally magnetized layer on the surface of the body whose strength at tZS is φ ; in other words, qua, external action, every lamellar magnet may be replaced by a complex shell on its surface.

There is, however, another way of looking at the result. Since A, B, C are derivable from a potential <∕>, the difference between the values of φ at any two points is simply the value of the line integral f (kdx + Udy + Cdz) along any path between those points. Hence if the tangential component of magnetization be given in direction and magnitude all over any surface, the value of φ, à constant près, is given all over that surface. We conclude therefore that, if a body be lamellarly magnetized, and we know the tangential component of its magnetization all over its surface, its external action is determined; for a constant c added to φ will simply add to the surface integral ffcosθdS c JJ B 2 ’

which, being c times the whole solid angle subtended by the surface at any external point, vanishes. For another very interesting proof of this result, see Thomson, Reprint, p. 398 sq.

Tl∣e vector potential of a lamellar magnet may be expressed by means of the formula [table]

These formulae furnish an immediate proof of the theorems of Thomson above stated.

By means of the last of them, it has been shown in the article Electricity vol. viii. p. 69. 2 that the vector potential of a simple magnetic shell can, as might be expected, be expressed by means of a line integral taken round its boundary; in the same place it has also been shown that the potential energy of such a shell in a magnetic field reduces to a similar line integral; and that the mutual potential energy of two such shells reduces to a double line integral taken round their boundaries.

An important approximate expression for the potential of a magnet at a point P, whose distance r from some chosen point in the magnet is great compared with the greatest linear dimension of the magnet, may be obtained as follows. Let the coordinates of P with respect to the chosen point and any axes through it be ξ, η, ζ; and let the coordinates of any point in the body referred to the same axes be x, y, z. Also let D — ∣ (ξ - αt) 2 +(ιj - y) 2 + (ζ-zj 2 ]∙ and let r = (ξ 2 + η 2 + £ 2 /, s=(x 2 + y i + ≈ 2 )^, t=ζx + ηy + ζz. Then the potential U at {x, y, z) of a unit pole placed at (ξ, η, ζ) is given by ττ 1 1 t 3f 2 -r¾ 2 5f 3 -3∕r⅜ 2 ιi u -D^T + ^ + -23- + -^- +1ic ∙

--l- + u 1 +tτ 1 + u,+⅛e..

by a well-known theorem, where U 1 , U 2 , U 3 , &c., are spherical harmonics of degrees 1, 2, 3, &c., in x, y, z, and - 2, - 3, - 4, &c., in ξ, V, C Now, by the theorem of mutual potential energy, the potential V of the magnet at (ξ, η, ζ) is the potential energy of the magnet in the field due to a unit pole at (ξ, η, £); hence by (17)

->J÷∙K>∣. . . <«), = V 1 + V 2 +V 3 + &c. '

where V 1 arises from U υ V 2 from U 2 , and so on. V 1 , V 2 , V 3 , &c., will be spherical harmonics in ξ, η, ζ of the most general kind, involving essentially 3, 5, 7, . . . 2t + l constants respectively, their degrees being -2, -3, . . . -i respectively. These constants will, however, depend in each case on a larger number of integrals taken throughout the magnetized body, thus the constants in V{ will depend upon ⅜(t + l)(i + 2) integrals. There is no diffi culty in writing down these terms except the length of the formula;. Putting fffkdv — lU, fffWiv—V.m, fffCdυ-'ikn ' ^-fffkxdv, M-fffKydv, ~X-fffCzdv - . (42), Ι ’ ≈fff ∖ Bz+Cy)dv, Q —&c., R —&c. J

we get

V α + ( 2L ~ m ~ y ½* + &c · + 3P√+ & c. + &c> (43 j.

It may be shown that, in the most general case, if we take tho axis of x parallel to the magnetic axis, and the origin at the point {(2L-M-N)∕2K, R∕K, Q∕κ}, and turn the axes about an angle tan ^ 1 P∕(M - N), the above reduces to v.S + i Mt,0 +4 , . . . (M) .

An interesting particular case is that in which the magnet is symmetrical with respect to the three coordinate planes. If wo take its axis to be in the axis of x, then, since all the integrals I., M, N, P, Q, R vanish, V 2 disappears, of the next set only

A 1 ∙=Jff^krfdv, k 3 ≈ffjkf t dv, k 3 ≈fffkz i dυ , (45) remain, and we get v Kξ, 3{A 1 (2ξ 2 -3√^3< 2 ) + A 2 (4^- C 2 -t 2 )+A√4C 2 -{ 2 - n 2 )}ξ v = r3 + 2?

(46).

The potential to the same degree of approximation of a positive and negative pole of strength μ, placed on the magnetic axis at distances + I. and - I. from the origin (centre of symmetry), is

2∕xL{ , 2∕zL 8 ξ(2ξ 2 -3η 2 -3< 2 ) v “r s + 2 r r

If we attempt now to find μ and I., so that the two magnetic systems may be equivalent, we find different values for I. for different positions of the external point. If, however, the magnet be symmetrical about its axis, so that A 2 = A 3 , then the expression for V reduces to

V K{, 3(A 1 -A 2 ½(2 { 2 -3 η 2 -3f), 47 .

v = r 3 + 2rτ - · - · wethenget2μL=K, and2μL 3 = 3(A 1 - A 2 ), whenceL 2 = 3(Aj - A s )∕K. In other words, in the case of a magnet which is symmetrical about its axis and also about an equatorial plane, we can represent the external action by means of a fixed ideal magnet, provided higher powers of the ratio of the greatest linear dimension of the magnet to the distance of the point considered than the fourth can be neglected. It is to be observed, however, that if A 1 <A 2 the length of the ideal representative magnet will be imaginary. ®

A convergent series for the mutual potential energy of two magnets M and M , may be obtained from the sextuple integral of (19). Let the origin be a fixed point 0 in M, and let the coordinates of a fixed point O ’ in M ’ with reference to a set of axes fixed in 0 be ξ, η, ζ; further, let x, y, z and a/, if, d be the coordinates of any elements dv and dv' in M and M ’, the axes being in the former case the system already indicated, in the latter a parallel system through O'; then, if r denote (ξ 2 + η 2 + ζ i ) i , an d δ 2 , δ 3 stand for l,i,l,we have dξ ’ dη dζ ’

W- -∕d√dι∕(Aδ 1 + Bδ 2 + Cδ 3 )(A'δ 1 + B'δ 2 + C'δ 3 )^l + Sw n ), where u n — 1 ∣ (a/ - z)δ 1 + (fy)δ a + (z z - z)δ 3 ∣ — ; n! ( > τ or

W-W 1 + W 2 + W 3 + (48), where W 1 , W 2 , W 3 , are spherical harmonice in ξ, n, ζ of degrees - 2, - 3, - 4, &c.

If we neglect all the terms except those of the first order, which amounts to supposing M and M' infinitely small, we get

W— -KK' ZZ'δ 1 2 + . . . +(mn ’ + m'n')δ. 2 δ 3 + . . · y ∙

If 0 1 , 0.> be the angles between the axes of M and M' and the line from the centre of M to the centre of M', and 0 12 the angle between the axes, this reduces to

W-5 *⅛os0 12 -3cos0 1 cos0 2 ) (49).

r i

From this formula we can derive at once by differentiation the force of translation and the couple about the centre of M', which represent the action of M upon it. An elegant synthesis of this action has been given for the most general case by Tait. It will be sufficient to confine ourselves here to the case where the magnetic axes are in one plane. In this case 0j 2 = 0 1 -0 2 , and W becomes KK' (sin0 1 sitf0 2 - 2cos0 1 cos0 2 )∕>∙ 3 . Denoting by X, Y, I. the forces of translation parallel to MM' and perpendicular to MM' (so as to decrease 0 1 ) and the couple tending to decrease 0 2 , we have [table]

One most important conclusion follows at once from these formulae, viz., that the translatory forces vary inversely as the fourth power of the distance, whereas the directive couple varies only inversely as the third power. Hence the couple may be quite sensible at distances for which the force of translation is inappreciably small. These conclusions apply of course equally to any pair of magnetized bodies, provided the distance between them be sufficiently great as compared with their linear dimensions. This, applied to the case of the earth, at once explains the phenomena that puzzled Norman and the earlier magnetic philosophers so greatly. The following particular cases are important (fig. 25)

(A) 0 1 = 0 2 = O, X=-^,γ = 0 , I. = 0.

(B) 0 1 = 0 2 = r, X = 3 -^, Y=0, I.-0.

(C) 0 1 = O, 0 2 = -, X = 0, Y = — K ' l- 2KK'

2 r 4 , ?· 3

(D) 0 1 ≡4, 0 2 = O, X = 0, Y, 3κκ ∖ I.-—.

2 r 4 ’ r s

The last two cases are especially important: the position of the deflecting magnet in (C) is described as “end on” (erster Hauptlage), in (D) as “broadside on” (zweiter Hauptlage); it will be noticed that the couple in the former case is double that in the latter.

If the terms of the second and third order be taken into account, and the magnet n's' be deflected through an angle φ from its original position by a deflecting magnet (I.) originally end on and (II.) originally broadside on, we get for the counles [table]

where T 1 and T' 1 are odd functions of the relative position of M and M', but T 2 and T' 2 are even. In the case where M and M' are symmetrical about three orthogonal planes, 0 and O' being the centres of symmetry, T 1 and T' 1 vanish, and the writer has obtained for the values of T 2 and T' 2

T 2 - - 6 { K(3A' 1 - 4 A' 2 + A' 3 ) - K'(2 A 1 - A 2 - A 3 )} 1

T' 2 = - ⅜{ K(12A' 1 -11 A' a - A' 3 ) - K'(3A 1 - 4 Λ 2 + A 3 )} ) where A 1 , A 2 , &c., have the meanings above assigned in (45). Sphere Magnetized in any Manner.— This is the most interesting of all the cases that fall under the present section, both from its being amenable to mathematical treatment and on account of its historical interest. It was first discussed in the beautiful memoir, entitled Allgemeine Theorie des Erdmagnetismus, in which Gauss laid the foundation of the rational theory of terrestrial magnetism. The following is a brief account of the theory, which has not been greatly added to since he left it.

Let X, Y, Z be the components of the earth’s resultant magnetic force at any point on its surface, in the directions of geographical north, geographical west, and vertically upwards respectively. The force is completely known when these are given, since it depends on three elements only. If H, δ, ι have the meanings formerly assigned (p. 220, 221, 227), we have of course

H = √X 2 + Y 9 , tauδ = Y∕X, tan∣ = Z∕√X 2 +Y 2 .

Again, if V be the magnetic potential of the earth» l the latitude, and λ the longitude of any point on its surface, then, supposing the earth to be a sphere of radius a, we have

X=-lf , Y=-_L_ Z ._^, . (53), α al αcost dκ dr r denoting the distance of any point from the centre of the earth. When V is known, therefore, the force is completely determined.

If now we suppose all the magnetized matter (or its equivalent—say, electric currents) to be within the earth, it follows, from the theory of spherical harmonics, that we can write down a convergent series for its potential at all external points, when the potential at every point of its surface is given. In fact, if the expansion of this surface potential in terms of surface harmonics be s 1 +s 2 + . . . +s l ∙ + . . . , we have for all external points

V-S 1 (⅜) ’ + ⅛(^) ’ ÷ · · · ÷<f + . . . (M). ·

The number of terms of this series that must be retained in order to obtain a sufficiently accurate representation of the phenomena will of course depend on circumstances, and can only be ascertained by trial. S 1 , S 2 , . . . S t ∙ are functions of known form, containing respectively 3, 5, . . . 27 + 1 constants; hence, if terms beyond the 7 th order may be neglected, the expression for V will contain 7 2 ÷27 arbitrary constants. These must be determined by observation, and then the magnetic action at all points on the surface or outside the earth is known irrespective of the internal distribution of the magnetic causes.

If we look at the matter from the general point of view that V is determined when its surface value is known, we have the following propositions.

I. V is determined when the vertical force is known at every point of the earth’s surface.

For, let the surface value of Z be expanded in a series of surface harmonics of which the 7 th is Z,∙; then, equating this to the ι th harmonic in the surface value of Z = - df∕dr derived from (54), we have (i + l)Si=αZ,∙, which determines S,∙. Thus the proposition is proved.

II. The surface value of V, and hence its general value for external points, is determined if the northward component of the magnetic force be known at every point of the earth’s surface.

This follows at once from the fact that the difference of the values of V at any two places is the line integral of the magnetic force along any fine joining them; thus, if V o be the value of V at the geographical north pole, we have

V = -af l Xdl + Υ 0 ...... (55).

∙∕ 7r

2

But the constant V o does not affect the general value of V; hence the proposition is established.

III. The same conclusion follows if the westward horizontal component be known all over the earth’s surface and the northward component along any one meridian.

In fact, if V be the potential at any place whose latitude is I and longitude λ, then N--af l Xdl-af κ X cos ldκ + V o ,

•Z 7r ∙∕ λo

2

the first integration being performed along the given meridian, the second along the parallel of latitude corresponding to the place.

From I., II., and III. we have the remarkable conclusion that, if the vertical component be given all over the earth, or the northward component, or the westward component and the northward along one parallel, then in each case the other two elements are determined.

Gauss gives another interesting application of the line integral of magnetic force. If this integral be taken all round any closed curve or polygon, the result is zero. Let us express this for any geodesic triangle ABC, at whose vertices the horizontal force has the values

H 1 , H 2 , H 3 . If the inclinations of H to BC at B and C be a and α', to CA at C and A β and β', to AB at A and B γ and y', then, if the arcs BC, CA, AB be not too long, we may replace the component along BC at every point by the average of its values at B and C, and so on. We thus get

⅜BC (H 2 cosa + H 3 cosα') + ⅛CA (H 3 cos0 + H 1 cosβ')

+ ⅛AB (H 1 cosγ + H 2 cos7') = O (56).

If we suppose the values of H at B and C to be known, and the values of the declination to be known at all three places, the above equation determines the value of H at A. Calculating in this way from observed values at Gottingen, Milan, and Paris, Gauss found for H at Paris 0’51696, the observed value being 0'51804.

It has been supposed hitherto that the magnetic causes are entirely internal to the earth. The foregoing theory enables us to test how far this assumption is correct.

If we suppose that there are external causes, then the potential at internal points due to these will be τ ∙ +τ *÷ +τ *(⅛) s ÷ · · τ '(÷)^

T o , T 1 , T 2 . . . T» being the different harmonics in the surface value of the part of the potential due to external causes. Suppose now the whole vertical force deduced from observation for all parts of the earth’s surface, and expanded in a series of surface harmonics, the i th of which is Z;; then, since this is the sum of the i th harmonics in the parts due to internal and to external causes, we have

-(i + l)S i + iT i =αZ i (57).

Further, suppose the surface value of V determined from observations of horizontal force, and let the i th harmonic in it be V t ∙, then we have

Si + T,∙=V i (58).

From equations (57) and (58) we can determine S,∙ and T i , and thus settle how much is due to external and how much to internal causes. It does not appear from observation that any sensible part of the mean value of V arises from causes external to the earth.

We have seen already that the acti<∙n of any body can be represented at external points by an ideal layer of positive and negative magnetism. Gauss finds for the surface density of the layer in the case of a spherical body like the earth, the expression (V/re - 2Z)∕4∙∏∙, which may be deduced immediately from the formulae already given.

If we draw a series of equipotential surfaces corresponding to small equidifferent values oï V, these will cut the earth’s surface in a series of equipotential lines, which are called the “magnetic parallels.” These lines obviously have the following properties. The horizontal force is everywhere perpendicular to them, and is at any point inversely proportional to the distance between two consecutive lines there. So that, if these lines were drawn upon a terrestrial globe, their crowding would indicate increase of horizontal force. The lines of horizontal force, or “magnetic meridians,” the tangent at every point of which is parallel to the horizontal component, are everywhere orthogonal to the magnetic parallels, and their positive direction is from p ιrallels of greater potential to parallels of less potential. If, as has been tacitly assumed hitherto in accordance with the results of observation, the potential on the earth’s surface have but one maximum and one minimum, then the parallels will be closed curves expanding successively from the maximum point and then closing again round the minimum point, and the magnetic meridians will all run between these two points. It is clear that at each of these points the equipotential surface and the earth’s surface touch; at the minimum point the line of total resultant force will pass to the earth, at the maximum point from it; at the former, therefore, the north end of a freely suspended needle will dip vertically downwards, at the latter the south end will do the same. This is the simplest possible case for a magnetized sphere. It is easy to see that, if we define a north pole as a point on the earth’s surface at which the horizontal intensity vanishes, and the dip is 90°, there might be more than one such point. Consider the series of equipotential surfaces 1, 2, 3, 4, 5, 6 in fig.

26, each of which has two eminences with a depression between them. The lines α, δ, c, d, e, f are the sections of these by the earth’s surface. 1 just touches the surface in a · and, if the potential increase in the order in which the surfaces are numbered, a will be a north pole. The section by 2 is the single oval b. 3 touches the surface in c', which is clearly another north pole, and also meets the surface in a single oval c equipotential with c'. The section by 4 is the double oval d, d'. The depression on 5 touches the surface at F, and meets it in a figure of 8, e, d, on which F is the double point. F is therefore yet another north pole according to our definition; it differs, however, from an ordinary north pole in one important respect; for the law that the north end of the compass points from parallel of greater to parallel of less potential shows at once that near F and inside the 8-shaped parallel the south end will point to F, whereas at a neighbouring point outside the north end will point to F. Such a point is called a false north pole, and we see that the existence of two true north poles necessitates the existence of a false north p.∙,le; and in general it may be established that, however many poles of the same kind there may be, true and false, the whole number must be odd. This of course disposes of the notion formerly held by some physicists that the earth actually had two north poles. As already indicated, Gauss concluded from his reduction of the magnetic observations at his disposal that, apart from purely local disturbances, the earth has, as a matter of fact, only one north and one south pole.

The effect of a deposit of magnetic ore, or other cause of the kind, might of course produce a disturbance, within a limited area, of the equipotential lines. It may assist the practical magnetist to indicate the nature of this disturbance in a particular case. Let us suppose that a magnet is placed some distance underground, vertical, with its north pole uppermost. Then, if its moment be sufficiently great, the equipotential lines will be as in fig. 27. The upper side of the figure is supposed to be magnetic north, and it is supposed that the undisturbed parallels would be straight lines running magnetic east and west, which is sufficiently near the truth in most cases. It should be observed that fig. 27 is in reality a transformation of figure 26, one of the poles being projected to infinity. The reader should notice that the double point F, due south of the point a vertically over the disturbing magnet, is a point of equilibrium at which the horizontal components of the forces of the earth and the magnet destroy each other; it will be a false pole, south or north according as the magnet or the earth prevails.

Experimental Foundation for the Law of the Inverse Square.

From what has already been laid down, it will be seen that the determination of the elementary law of magnetic action is a very complex problem. The action between two magnets depends, not only on their distance apart, but also on their relative angular position. Then we have to distinguish force of translation, which varies inversely as the fourth power of the distance, and directive couple, which varies inversely as the third power. It must also be remembered that the elementary law results in part from an hypothesis as to the nature and distribution of the cause of the magnetic action, for, until some such hypothesis is made, no clear conception is possible of what is to be understood by elementary action. Lastly, we have the disturbance which arises from magnetic induction, the consequence of which is that magnetically speaking two magnets are not the same at different distances apart. When all these circumstances are considered, it is not surprising to find considerable uncertainty and difference of opinion among the earlier magnetic philosophers. The truth is that the law as now established owes quite as much to the development of magnetic theory as to the work of magnetic experimenters.

The question attracted the notice of Huygens and Hooke, but Newton seems to have been one of the first who propounded any law on the subject. He says {Principia, lib. iii. prop. 6, cor. 5) that some rough experiments had led him to the conclusion that the magnetic force (vis magnetica) decreases according to the ]aλv of the inverse cube of the distance. No account of the experiments is extant, and it does not appear what he means exactly by vis magnetica. If the directive couple is meant, and the action of the entire magnet is intended, then, as we have seen, this is in agreement with modern theory. In a remarkable note in the annotated edition of the Principia by Le Sueur and Jacquier (assisted by Calandrini T) (1742) on the passage inquestion, a series of deflexion experiments are described, and an accurate discussion is given, from which results the law of the inverse cube for the deflecting couple. Hawksbee made experiments with a view to determine the law of magnetic action, in which a deflecting magnet was moved at various distances round a compass, and the corresponding deflexions noted. A few years later Brook Taylor and the same experimenter made a series of observations in which the “end on” method of deflexion still in use was adopted. But in neither case was any definite result arrived at. A similar uncertainty appeared in the experiments of Whiston, who indicates the inverse f th power of the distance as the law of decrease. Musschenbroek’s experiments, which were extensive, also led to no final result. He used the method of Hooke, in which the attraction of a vertical bar magnet upon another suspended from one arm of a delicate balance is balanced by weights attached to the other arm. F rom some of his experiments he deduces as low a power as the inverse 1st, from others the ∣th, and so on; but no attempt is made to analyse the phenomena. Michell, in his treatise on artificial magnets (1750), however, deduces the law of the inverse square from Musschenbroek’s results. Although Aepinus does not arrive at any definite result as to the elementary law, there can be no doubt that his Tentamen Theoriae Electricitatis et Magneiismi (1759) contributed powerfully towards the solution of the question. Tobias Mayer seems to have been the first to publish the law of the inverse square as the actual result of an experimental investigation. His paper was read before the Royal Society of Göttingen, and was referred to in the Göttinger Gelehrter Anzeiger for 1760, but never fully published; it is best known from the criticism of Aepinus, “Examen Theoriae Magneticae a Tob. Mayero propositae” {Nov. Comm. Acad. Petrop., 1768). The most important of the earlier contributions was undoubtedly that of Lambert. He seems to have been the first to analyse the physical circumstances of the problem in a thorough manner, and to point out the various elements of disturbance to be provided for. We regret that we are unable to devote space to an exhaustive account of his memoirs, which are most instructive reading even now. He showed that the effect of an oblique magnetic force on the needle varies as the sine of the inclination; and, making allowances for this, he deduced the law of the inverse square from deflexion experiments made at different distances. He also described the method of oscillations, but found difficulties in its practical application. It is upon his theoretical work, however, rather than upon his experiments, that his claim to be remembered rests. About the same time as Lambert, we have Dalla Bella and Robison, the well-known professor of natural philosophy in the university of Edinburgh, working at the same subject. The former used the method of Hooke and Musschenbroek, but discussed more carefully the exact nature of the resultant action. His results indicated the law of the inverse square. Robison used both the method of deflexion and the method of oscillation, the peculiarity in his apparatus being the movable magnet, which was composed of two magnetized spheres connected by a slender rod, and suspended either in the field of the earth alone, or at different distances from a large magnet. He made several independent investigations, and seems to have arrived in each case at the law of the inverse square as his final result.

The researches of Coulomb, from which many date the commencement of the modern theory, present many features of great interest. He used the improved form of Michell’s torsion balance, which had served him so well in his electrical experiments. In order to realize as nearly as possible the ideal case of a linear solenoid, whose action can be represented by positive and negative magnetism concentrated at its ends, he worked with magnets made of thin steel wire magnetized longitudinally. The circumstances of the experiment are thus considerably simplified, for the acting magnet may be so arranged that the action of one of the poles may be neglected, or, failing that, the action of both can be easily calculated.

In one of his experiments he took a magnetized steel wire 25 inches long, and 1⅜ lines thick, and placed it vertically in the magnetic meridian before a horizontal magnetic needle some 3 inches long, delicately suspended by a silk fibre. The rod was raised and lowered at a given distance from the needle until the attraction on the near pole of the needle, as tested by the rapidity of the vibrations, was a maximum; it was then found that the lower end of the bar was about 1 inch below the needle. Again, the rod being placed horizontal and perpendicular to the magnetic meridian on a level with the needle, it was displaced until the needle returned to the magnetic meridian; it was then found that the needle was directed to a point about 1 inch from the end of the bar. Both these experiments thus indicate that the magnetism at one end may be supposed concentrated at a point about an inch from the end of the bar. It is clear that, in these experiments, provided the rod is sufficiently long or the distance between it and the needle not too great, the action of the distant pole may be neglected, for the double reason that the pole is more distant and that the force exerted by it is nearly perpendicular to the direction in which it can be effective, ∆iaking this assumption, Coulomb observed the number of vibrations, when the vertical rod was absent, and when it was placed at various distances.

The forces thence deduced were found to vary very nearly as the inverse square of the distance. Statical experiments with the torsion balance led to a like result.

Later than Coulomb we have the experiments of Bidone, Hansteen, Steinhäuser, and Scoresby. By far the most important among these is Hansteen, whose methods were a great step towards the more complete treatment finally adopted by Gauss. He uses Taylor’s “end on” method of deflexion, and also the method of Hooke and Musschenbroek. The acting magnet was a bar magnet, the action of which he represents by a distribution of positive and negative magnetism on its two halves whose density at a distance x from the centre is λx r . The force at distance D due to an element dp of positive magnetism he assumes to be tfyc∕D n . He finds that in all his experiments the value n = 2 best represents the results obtained; but that various values of r may be adopted with almost equal advantage; he inclines, however, to the value r = 2.

In his classical memoir on the absolute measurement of the earth’s magnetic force, Gauss took up the question in < the most general manner yet attempted. Assuming that the force due to an element of positive magnetism varies as the inverse nth power of the distance, he showed that, when the distance between the magnets is sufficiently great compared with the greatest linear dimensions of either (more than four times as great in his own experiments), the deflexions ≠ and φ' for the “end on” and “broadside on” positions of the deflecting magnet are given by tan<∕> =I. l τ∙ - < n + 1 ) + I. 3 r^<' t + 2 )+ &c., tan φ' =I. 1 'r * < n + 1 > + I. 3 'r - <"+ 2 > + &c.; where I. 1 ∕I.' 1 = n. He made a series of deflexion experiments, and found that his results could be represented with sufficient accuracy by the formulae tan≠ =0O86870r- 3 -0∙002185r- 5 , tan≠'=0O43435r- 3 + 0O02449τ- 5 .

The following table shows the closeness of the agreement between theory and experiment (r is measured in metres j Φ and Φ' denote observed and φ and φ' calculated values):— [table]

We have here a double proof of the law of the inverse square,—first, in the fact that tan<∕> and tan<∕>' can be expressed so accurately by two terms of a series, the first of which contains r^ 3 ; second, in the fact that the coefficient of the first term in tan<∕> is exactly double that in tanφ'. These researches of Gauss are remarkable, not only for the great generality of the theory, but also for the novelty of the experimental method, and the exceeding accuracy and refinement of the observations. The law of the inverse square has in fact been regarded as settled ever since they were made. They are important from another point of view, to which we shall return presently.

Magnetic Measurements, Relative and Absolute. he mos. important magnetic determinations that have to be made are the direction of the axis of a magnet relatively to its mass, the magnetic moment of a magnet, the direction of a magnetic field, and the strength of a magnetic field, or its component in any given direction. In most of these cases the measurement may be either relative or absolute. For example, we may determine the moment of a magnet either relatively, in terms of the moment of some other magnet arbitrarily chosen, or absolutely, in terms of the fundamental units of space, mass, and time. The complete theory of measurements of the latter kind is due to Gauss, and the carrying of them into practice to him in conjunction with Weber and the Magnetischer Verein, of which these two German philosophers were the leading spirits. We shall discuss the matter here only in so far as it concerns the work of a physical laboratory, the rest belonging more properly to the subject of Terrestrial Magnetism (see Meteorology). Axial Direction and Magnetic Declination.— The magnet is suspended, usually by means of one or more fibres of unspun silk, so as to be free to move about a vertical axis. We shall suppose, for simplicity, that the magnetic axis is in a horizontal plane. If this is not so, instead of determining the axial direction, we determine a vertical plane through it. In order to obtain a fixed line of reference in the magnet, two marks may be made on it as nearly in the direction of the axis as can be guessed to begin with; this arrangement is used with dipping needles and also for horizontal needles when no great accuracy is required. For declination needles two contrivances of greater refinement are used.

1. A mirror is” rigidly attached to the magnet, so that the normal to its surface is nearly parallel to the magnetic axis. The image of a fixed horizontal scale in this mirror is observed by means of a fixed telescope, and the angular motion of the magnet deduced from the motion of the scale divisions over the wires of the telescope. This is called the mirror method.

2. A more compact arrangement is to attach to the magnet a small photographic scale and a lens, the former being placed at the principal focus of the latter, so that the line joining the middle division of the scale to the optical centre of the lens is nearly parallel to the axis of the magnet. The scale is viewed through the lens by means of a fixed telescope, and it is clear that the line just mentioned gives us a fixed direction in the magnet, and that the motion of the magnet can be followed by observing the apparent motion of the scale across the wires of the telescope. This may be called the collimator method.

The apparatus usually employed in the United Kingdom for observing the magnetic declination, and also for other absolute magnetic measurements, is the portable unifilar magnetometer, the upper part of which is shown in figs. 28, 29. The lower part consists simply of a tripod stand supporting three V-shaped groves, into which the points of the levelling screws attached to the fixed limb of the instrument are set. In making an observation of the declination the instrument is arranged as in fig. 28. The declination collimator magnet is suspended in the box A (the sides of which are removed to allow the interior to be seen) by means of the suspension fibre D, attached to the torsion head PH. The scale of the magnet is observed through the small telescope QBG. The first step is to remove the torsion as far as possible from the suspension fibre by hanging to it a brass plummet E of the same weight as the declination magnet. After this weight has come to rest, it is replaced by the declination magnet, so that the latter shall rest as nearly as possible in the magnetic meridian without introducing torsion of the fibre. The movable limb is now turned till some division of the magnet scale is on the cross wires of the telescope. It is then clamped. The magnet is now inverted, and the number of the scale division on the wires again read. The mean of these readings gives the point of the scale the line from which to the centre of the collimating lens is parallel to the axis of the magnet. This point of the scale (axial point) will remain the same so long as the magnetism of the magnet does not alter, or the adjustment of the scale and collimating lens is not interfered with. The tangent screw is now worked until the cross wires of the telescope are on the axial point of the scale. The verniers of the limb are then read. The next step is to observe the azimuth of the sun or of some other heavenly body, by means of which we can refer the azimuth of the magnetic meridian to the true north. For this purpose the instrument is provided with a small transit mirror NO, which has a motion in altitude so as to bring any object into the field of the telescope. To use it, the limb is unclamped, and it and the mirror moved until the sun or star comes into the field of the telescope; the limb is then clamped and the time noted at which the heavenly body passes the intersection of the cross wires. The verniers are again read. The differences of the readings, added to the azimuth of the heavenly body found by means of the time from the Nautical Almanac, gives the declination at the time and place of observation.

There are several causes of error to be guarded against. (1) Torsion is reduced within as small limits as possible to begin with, and if there is reason to suspect any residual error we may test the apparatus by turning the torsion head of the suspension tube first 90° one way and then 90° the other. If the deflexion of the magnet is exactly the same and oppositely directed in the two cases, then we may conclude that the torsion is zero in the azimuth of equilibrium. If not, then we may turn the torsion head so as to reduce the error still further; or we may calculate its amount (assuming torsion to be proportional to twist) from the two observations, and allow for it. (2) If the axis of the magnet is not very nearly parallel to the line of collimation of the telescope to begin with, and consequently the two scale readings far apart, an error may arise from the vertical axis of suspension not being exactly reversed by the inversion. This error is reduced by repeating the observation, after adjusting the axis of the magnet and telescope so as to be more nearly parallel. (3) If mean declination for a given day be desired, correction must be made for the diurnal variation, and under certain circumstances this variation may even produce disturbances in the course of a single observation.

Magnetic Moment, Horizontal Intensity of the Earth’s Force.— If merely relative measurements of the magnetic moment K of a given magnet, or of the horizontal intensity H of the earth’s force, are desired, there are two methods of obtaining them. The first is the method of vibrations. Having found the moment of inertia A of the magnet M about its vertical axis of suspension, and the time T of its vibration under the earth’s force, we obtain the product KH, — p, say. Secondly, by the method of deflexion, of which two varieties, tangent deflexion and sine deflexion, are in use, the value of the quotient K∕H, =q, say, is found. In this method K is used as the deflecting magnet, and the moment K' of the deflected magnet does not appear in the result. It is obvious that, if we know the value of H, or may assume it constant, either of these methods will enable us to express the moment of any magnet in terms of that of another arbitrarily chosen as unit; and, reciprocally, if we operate with a magnet of known or of constant moment, we can determine the values of H at different times and places in terms of its value at an arbitrarily chosen time and place.

By combining two observations, in one of which a magnet K is the vibrating and in the other the deflecting magnet, we can obtain both K and H in absolute measure, for we have two equations KH =y>, and K/H = 7, which give _

K= "dρg, and H = ∖ ∕p ∣ q. Vibration Experiments.— If θ be the angle between the axis of the magnet and H at time t, y the angle between the axis and H in the position of no torsion, τKH the coefficient of torsion, them the equation of motion of the magnet, when the arc of oscillation is very small, may be written

A0+KHβ + <rKH(0- 7 )-O (59).

This gives for the period of a complete vibration

T 2 =4π 2 A∕KH(l +τ) (60).

The observations are made with the magnetometer arranged as for the declination experiment. The swinging magnet is brought to rest, and the circle so clamped that the axial point of the magnet scale is on the cross wire of the telescope; the magnet is then slightly disturbed so as to oscillate through a small arc (16' or so). The time of vibration is found first roughly, by taking the time of a single vibration, then more accurately by counting a large number of vibrations and timing the end of the last as accurately as possible. τ is found by observing the deflexion θ' and θ" caused by turning the torsion head through an angle β in one direction and then through an angle β in the opposite direction; we thus get from equation (59)

KHθ' + τKH(0' - y — β y ) = 0 , KH0'' + tKH(0"- 7 + 0) = Oj and τ = (0' -θ") ∣ (2β-θ ’ + 0"). From the same equation we may also determine 7 when necessary.

The most troublesome part of the whole process still remains, viz., the determination of A. This is effected by attaching to the magnet a body whose moment of inertia B can be calculated from its dimensions. For this purpose Gauss fixed a cross bar of wood to the magnet, and attached to it at known equal distances from the axis of suspension two cylindrical weights of known mass and dimension. Sometimes a cylinder of gun metal is slung below the magnet by means of two loops. Perhaps the best method is to use a ring of gun metal attached to the magnet so that its plane is horizontal and its centre as nearly as possible in the line of suspension. The new time of vibration being T 1 , and the new coefficient of torsion (if different) τ', we have the new equation

T 1 2 =4τ 2 (A + B)∕KH(l+√).

From this and (60) we get -b∕(⅛∙¾ , -∙)∙

There are several corrections which, although in general negligible may sometimes require to be considered. (1) H may vary so much during the experiment as to cause a sensible error; (2) if the arc of vibration be too large, it may be necessarv to apply the reduction to infinitely small arcs; (3) if the amplitude of the vibrations decrease too rapidly, account must be taken of the resistance to the motion arising from the viscosity of the air, &c.; (4) a correction has to be rna.de for the alteration of the moment of the magnet by the earth’s induction, and (5) a temperature correction for the magnetic moment and the moment of inertia.

Deflexion Experiments.—In Gauss’s arrangement the deflecting magnet was placed in an east-west direction, r.e., end on to the original position of the deflected magnet. The equation of equilibrium in this case is [see equation (51)]

KΗ(l+τ)sin0=cose(^y^-il+^∣-∣- . . . )

∖ r i r i r 5 J or r,5 ⅛ 1,ta,1, - 1 -r + ⅞ + < βl >∙

where P 1 = T 1 ∕2K'K, P 2 = T 2 ∕2K'K, &c.

In the method of sines the deflecting magnet is turned until it is perpendicular to the axis of the deflected magnet in its final position of equilibrium; the equation of equilibrium in this case is r 3TT p p

- stoβ .ι-i + ^ +tc (e2) .

The advantage oΓthe method of tangents is that the moment of the deflector is not affected inductively by the earth’s force. In the method of sines a Correction has on this account to be made; but, on the other hand, there is no torsion, and, from the symmetry of the position of the two magnets, the approximate formulae have a more exact application.

The new pattern of the unifilar magnetometer is adapted for the method of sines. The instrument arranged as in fig. 29 is first carefully levelled, and fitted with the graduated cross bar D, which is so set in its sockets as to be perpendicular to the line of collimation of the telescope A. The box is opened, and the torsion removed from the suspending fibre by means of a plummet as already explained. The deflected magnet is then suspended so as to be at the same height as the deflecting magnet wh< n the latter is placed in its carriage on the cross bar. The sides of the box are now closed, and the circle of the instrument turned until the middle division of the scale B, seen by reflexion from a mirror attached to the deflected magnet, is on the cross wires of the telescope A; the circle is then clamped, and the verniers read. The deflecting magnet K (the same as that need in the vibration experiments) is next placed in its carriage I. on the cross bar at a distance r 1 (30 cm. or so) east; the circle is then turned until the middle division of the scale is again on the cross wires; the verniers are read once more. Ιhe difference between the two readings being θ 1 , we have 7∙ 1 3 Hsine 1 U l ⅞ + (6 3).

2 K r 1 r 1 2

The deflecting magnet is reversed in its carriage, and the whole operation repeated. If the deflexion now be θ 2 , irrespective of sign, then

^ si n¾ = 1+ S +j ⅞ + (64).

The mean of these gives

' ⅞^(sin0 ι + sin0 a ) = l+⅞+ . . ’. . . (65).

The magnet is finally removed to a distance r l west, and the previous observations repeated; we thus get

(sin θ 3 + sin θ 4 ) = 1 + ⅞ + (66).

The mean of (65) and (66) is then taken, and we get

⅛Γ≡ 1 ≡ 1 ÷⅜ (67) ’

where S 1 = ∣ (sin θ 1 + sin θ. 2 + sin θ 3 + sin 0 4 ), or, what is practically the same, the sine of the mean of 0 1 , θ 2 , θ 3 , and θ 4 . The object in taking the mean of (65) and (66) is to eliminate any error arising from the non-coincidence of the middle point of the cross bar with the axis of suspension.

In order to eliminate P 2 , another set of observations are made with a new distance r a (26 cm. or so), giving the equation

⅛F s ∙- ι+ ⅞ < β8 >∙

From (67) and (68) we have finally

K r 1 δ S 1 - r 2 δ S 2

H 2(r 1 2 -r a 2 ) ’

v WwS 2 -r 1 3 S 1 ) 2 ^ r 1 δ S 1 -r i δ S 2 *

When great accuracy is required, several corrections have to be applied:—(1) the moment of the deflector must be corrected for induction; (2) the moment of the deflector must be corrected for temperature; (3) the lengths r 1 and r 2 on the cross bar must be corrected for temperature.

Statical Method.— There is another method by which we may determine the product KH, viz., we may oppose a statical couple to the couple exerted by the earth on the magnet in a given position, so that there may be equilibrium; the statical couple, which may arise from the torsion of a fibre, from a bifilar suspension, or other gravitational force, thus becomes the measure of the magnetic couple; and hence KH can be determined in absolute measure. Coulomb’s torsion balance experiments are an example of this method. It finds numerous applications in the variation instruments of fixed magnetical observatories, and also in instruments for magnetic observations at sea, but it is very little used in the ordinary work of a physical laboratory.

Magnetic Measurement by Electromagnetic Induction.— It has been explained in the article Electricity that, if, either owing to the variation of the magnetic field, or owing to the motion of a closed linear conductor in it, the number of lines of magnetic force N passing in the positive direction through the conductor vary, this variation will cause an electromotive force - dN∕<Zi in the positive direction round the circuit. Let us suppose, to take a simple case, that we have a coil of wire made up of a number of parallel plane circular windings, and that the sum of all the areas of the separate windings is A. If we place this in a field of uniform intensity R, so that the normal to the windings makes an angle Θ with R, the number of lines of force passing through the coil will be N 1 = ARcos0. If we now suddenly reverse the coil, by turning it through 180° about an axis perpendicular to its normal, the value of N in the new position is N 2 = -ARcosØ. Hence the integral electromotive force during the motion is - yd‰7N∕c⅛ = N 2 -N 1 = - 2ARcos0, and the whole quantity Q of electricity which passes will be Q = - 2ARcos0∕S, where S is the resistance of the coil. If Q be found in absolute measure, and A and S be known, we thus obtain the value of RcosØ. This is the principle of Weber’s “earth inductor,” by means of which the horizontal and vertical components of the earth’s force can be measured, and in consequence the declination and inclination determined.

If the test coil be made very small, so that the portion of the field which it occupies may be supposed uniform, this method may be applied to measure the intensity at different parts of a non-uniform field. The small coil is placed with its windings perpendicular to the lines of force, and then suddenly reversed, or, if that be impossible, suddenly removed to a part of the field where the number of lines of force passing through it is zero. The integral electromotive force is of course in the latter case only half what it is in the former. This method is often of use where, owing to the great strength of the field and the consequent disturbances arising from induction, any other method would be utterly useless.

The method of electromagnetic induction may also be applied to measure the component of the magnetic moment of any body parallel to a given line.

Let αα ’ bb' (fig. 30) be the section of a uniform cylindrical coil of length 2l, made up of a single layer of flat circular windings of radius b, n to the centimetre. Let the axis of the coil be taken for a>axis, and let K be any magnet within the coil, placed with the given line parallel to the axis of the coil. Let pq be any single winding of the coil, then the surface integral of the magnetic induction for pq is given by ffadydz ; hence the whole number of lines of force through the coil is given by

N = fndxjfadydz, =n fjfadxdydz, the integration being extended all over the cylindrical space abb'a'. Now, since α=α + 4τrA = -dN ∣ dx + iπh, we get

N = - *^^^χdydz + ^πnjJ^Mlxdydz, ^ ∣ = — n(JJNdydz— JJTdydzj + 4τrwK ,

= + 4πnK — n(S — S') ∙l where K is the component parallel to the axis of the coil of the moment of the magnet, and S and S' the values of the surface integral of the potential of the magnet (derived from Poisson’s distribution) over the two ends of the coil. When there are more layers than one, we must of course sum the different parts of N arising from the different layers.

The formulae are quite general, and some applications will be given later. Meantime we see that, if the coil be so long that the magnetic potential of the body at its two ends may be neglected, then the integral electromotive force caused by the sudden removal of the body, or by the sudden destruction of its magnetism, is 4τrn times the component of the magnetic moment parallel to the axis of the coil, n being the number of windings per unit of length of the coil.

Historical Remarks on the Progress of Magnetic Measurements.— The method of vibrations came very early into use iu magnetic measurements. Whiston and Graham made vibration observations with a dipping needle. Musschenbroek and Mallet also used a horizontal needle. Lambert appears, however, to have been the first to thoroughly understand and appreciate the method. For long it was the only accurate process in use for obtaining relative measures of the earth’s force. It was so used by Rossel, D’Entrecasteaux, and Humboldt. Coulomb, Hansteen, and Poisson, all contributed more or less to its improvement; and it finally reached perfection in the hands of Gauss, who gave the experimental process for obtaining the moment of inertia, investigated the correction for resistance, and, by the introduction of the mirror and scale method, imparted astronomical accuracy to the determination of the period of vibration.

The method of deflexion, in one form or another, is very old. Its existence as a thoroughly scientific method, however, dates from Hansteen. The essential improvement of eliminating the constants depending on the magnetic distribution by observations at different distances is due to Gauss. The advantages of the sine method were first pointed out by Lamont in 1841.

Poisson seems to have been the first to conceive the idea of absolute magnetic measurement In a short but luminous article at the end of the Connaissance des Temps for 1828, he describes a method for obtaining the value of H in absolute measure. Horizontal vibration experiments are to be made with two magnets A and A', whose moments of inertia A and A' are known. The times of vibration t and t' of A and A', each suspended alone, are to be observed. Then both are to be placed in the magnetic meridian at a distance r apart in the same horizontal line, and the periods θ and θ' observed, of A when A , is fixed, and of A' when A is fixed. If r be very great compared with the linear dimensions of A and A', then jp_ 8π 2 00'∖∕AA'

“r 3 w√(i β -0 2 Xt' i -0' 2 )'

He recommends, however, that comparatively small values of r be taken, and the constants of distribution eliminated by experimenting at different distances. His fundamental units are the gramme, metre, and second.

Nothing came of Poisson’s proposal until Gauss took up the subject, both theoretically and experimentally, as above described. The first absolute measure of the earth’s horizontal force was made by him at Göttingen on the 18th September 1832; the value found was 1 ’ 782 in millimetre milligramme second units. The magnet he used (about a foot long and weighing about 1 fl>) had for its moment 100877000 in the same units.

The determination of the distribution of magnetism within ’ a body, in other words, the determination of the magnetic moments of its individual elements, by observations of magnetic force at external points, is, as we have seen, an indeterminate problem. Nevertheless, a considerable part of the literature of magnetic science relates to it; and we must give some account of what has been done, although the results obtained are of comparatively slight physical interest, and of small practical value.

Experimenters have been somewhat slow in recognizing the essential indeterminateness of the problem. This no doubt has arisen from their imperfect analysis of the phenomena. Thus, although we cannot determine the actual internal distribution, yet the problem to determine the Gaussian surface distribution which will represent the magnetic action at all external points, however difficult, is quite determinate. This surface distribution has been called by some the “free magnetism” of the body; and some, all the powerful contrary evidence notwithstanding, have imagined that this distribution has a physical existence, and have even spoken of the depth to which the free magnetism penetrates into the magnet. Others have confounded the free magnetism of Gauss’s distribution with that of Poisson’s; and in many cases it is impossible to gather what the experimenter meant to indicate exactly by the phrase.

The case in which, from the circumstances, the variation of the internal distribution is confined within the narrowest limits is that of bar magnets, whose length considerably exceeds their lateral dimensions; and this is practically the only case that has been much studied. The most natural way of attempting to represent the action of such a magnet would be to suppose it replaceable by a fixed ideal magnet, and then to determine by experiment the strength and position of the poles of this magnet. The earliest notion was that the poles were situated exactly at the ends of the bar. It was soon found, however, that, if the poles did exist, they were not in general exactly at the ends. Lambert and Kupfer concluded from their experiments that in many cases the poles lay outside the bar, while in weak magnets they lay inside. Coulomb, as we have seen, and also Dalla Bella, inferred from their results that the poles fell within. Recent experiments have been made by Pouillet, by Benoit, by Petruscheffsky, and by others on the same subject; but it is needless to describe them here.

The word “pole,” like the phrase “free magnetism,” has been used by different writers in very different senses. Some have applied that name to the mass centres of the positive and negative magnetism of the actual molecules. But, although as a matter of convenience we have used these points in our theoretical development, they have, as far as physical observations are concerned, no existence. Others have defined the poles to be the mass centres of the positive and negative parts of Gauss’s surface distribution. These might of course be determined, although the process would be extremely troublesome, and the result of no practical value whatever. In point of fact, if the magnet be in a uniform field, i.e., at a very great distance from the system that acts on it, the action depends solely on the magnetic moment, and the magnetic distribution has nothing to do with it; the poles in this case are physically indeterminate. If, on the other hand, two magnets are within a moderate distance of each other, we may set to ourselves the problem to find two points in each of them such that the mutual action will be represented by quantities of positive and negative magnetism concentrated there. Then, in general, such points may or may not exist. Riecke has shown (see above, p. 233) that, if the distance between the magnets exceed a certain limit, then, as a matter of approximation, these equivalent poles, as he qalls them, do exist. Except, however, in the case of magnets symmetrical about an axis, and also about an equatorial plane, they are not fixed in the magnets, but depend upon their relative position. Although his results are extremely interesting from a mathematical and theoretical point of view, we do not see that much practical advantage would attend the use of these equivalent poles; and we are inclined to think that, except in the popular usage for distinguishing one end of a magnet from the other, and in the case of ideal magnets, the word pole had better be abandoned altogether.

The idea of representing the action of a linear magnet by a continuous distribution of free magnetism, positive in one half and negative in the other, is very old. It appears in Bazin’s work on the magnetic curves published in 1753; and Tobias Mayer, in his memoir above quoted, assumes that the density of the distribution is proportional to the distance from the middle of the bar. Four distinct methods have been used in attempting to determine the law of distribution.

1. The deflexions of a small needle in different positions near the magnet have been observed, and by means of these the constants in some formula assumed for the distribution have been calculated. This was the process adopted by Lambert and Hansteen, and, in some of his experiments, by Lamont.

2. Instead of measuring deflexion, we may count the oscillations of the needle, and proceed as before. This method was used by Coulomb, Becquerel, and Kupfer, but it led to no satisfactory results, partly owing to the disturbances arising from induction and the force of translation upon the needle, partly owing to the difficulty of putting a satisfactory theoretical interpretation upon the results.

3. Some observers have measured the force required to detach a small armature of soft iron or steel from different parts of the bar, thinking thereby to obtain a direct measure of the free magnetism. It is not very easy to say what is measured by this process, but it is obvious, on a little consideration, that the effect is complex, depending greatly on the nature and extent of the surfaces in contact, and also upon the mutual induction between the magnet and the armature. Experiments of this kind have been made by Dub, Lamont, and others.

4. Another method frequently employed is to slide along the bar a small ring-shaped coil embracing it as closely as possible, and to measure the induction currents for a given displacement. The assumption usually made is that the integral electromotive force is proportional to the free magnetism on the portion of the bar passed over, or, what amounts to the same thing, to the difference between the magnetic moments per unit of length of the sections of the bar on which the coil rests at the beginning and end of the motion. This is, however, only an approximation to the truth, and the accuracy of this approximation is very difficult to estimate in the practical case where the lateral dimensions of the bar are finite. The following investigation will show the nature of the difficulty.

The integral electromotive force is - (N 1 - N 2 ), where N 1 and N 2 are the surface integrals of magnetic induction taken over the coil in its initial and final positions. Let us take first a linear solenoid SN (fig. 31) of length l, and magnetic moment m, and a coil of a single winding PQ, which moves so that its centre R is always in the line SN, and its plane always perpendicular to SN; then

N = ffτdydz =ffadydz + ±ιjfhdydz, the former integral extending all over PQ, the latter over the infinitely small section of the solenoid at R, « being the force due to the end distribution at N and S. We thus get

N=~(cos0- cosfl') (70),

V 9 where θ and θ' are the angles PSX and PNX. This shows, in the first place (see equation (2δ) above), that if the coil PQ were to expand and contract as it moves, so as always to remain a section of the same tube of force, there would be no variation of N, and no electromotive force, which is as it should be. If we were at liberty to suppose PQ infinitely small, then, when R is between S and N, cos θ - cos fl' would be the sum of two unities, and, when R is outside, the difference. In such a case, so long as PQ moved on the magnet, there would be no electromotive force, but if we suddenly move it over the end, there would be an electromotive force - iιrm∕l, which is proportional to the moment of the bar. When PQ is not infinitely small, there is a variable part of N, depending on the dimensions of PQ, which will give rise to an electromotive force, even when the coil is moved along a uniformly magnetized bar, where there is no free magnetism except at the ends.

It is now easy to form a conception of what happens in the case of an ordinary complex solenoidal bar. We may suppose such a bar made up of a number of simple linear solenoids. A certain number of these, corresponding to the end parts of Poisson’s distribution, will have the same length as the bar; the others, corresponding to the lateral surface and volume parts of the distribution, will be of continuously diminishing lengths. If we were at liberty to suppose the lateral dimensions of the bar and the radius of the coil to be infinitely small, then, as the coil moves along the bar, we should have an electromotive force due to passage over the ends of the short solenoids, and, as it moves over the end, an electromotive force due to passage over the ends of the long solenoids. We might in this way by a sufficient number of observations determine the distribution of the free magnetism throughout the bar and at its ends; and in this case no distinction would be necessary between the volume and the surface, distribution in any section.

If, however, the dimensions of the section of the bar, and consequently of the coil, be finite, a correction would have to be applied, depending, not only on the dimensions of the bar and coil, but also on the magnetic distribution. All that we can then do is to assume a formula for Gauss’s surface distribution and determine its constants. We thus get Gauss’s distribution, and a formula that will account for the electrical observations; but we obtain no information as to the actual internal distribution of the magnetism in the bar.

Lenz and Jacobi appear to have been the first to apply the method of induction currents to the measurement of the magnetic distribution in bar magnets. They attempted no theoretical analysis of their results, although they assigned a law of distribution. Van Rees, who questioned their conclusions, gave an imperfect theory, and made some careful researches of his own. Rothlauf made further experiments, and entered more fully into the theory, though still with insufficient generality. The most recent experiments of the kind we are aware of are those of Schaper, who discusses the theory with complete generality, taking account of the ends of the bar.

After what has been said, the reader will scarcely be surprised to find that the different experimenters assigned very different formulae for the distribution in bar magnets. Lambert deduces from his experiments a distribution whose density is Air,—A being a constant, and x the distance from the ends of the bar. Brugmans, V. S winden, and Lenz and Jacobi adopt the law Air 2 ; Hansteen, as we have seen, the law Aaf, where r = 2 or 3. Biot deduced from Coulomb’s experiments the law A(μ,*- p~ x ) for the density of the free magnetism, which would give for the moment per unit of length of the bar the law a - l>(a x + μ~ x ), see above, p. 231. Becquerel, Van Rees, Lamont, and Rothlauf favour this last formula; but none of these experimenters give any proper account of the ends, which must be specially represented in all but those cases where the magnetic moment is zero there. Schaper finds that the results of experiment can be adequately represented by means of end distributions, and a lateral surface distribution following the law kx + Ba? 3 . See his paper above quoted, p. 242. 1 Carrying Power of a Magnet.- —It is obvious that the magnetization of a piece of iron must affect its force of cohesion. The most familiar case is that of a magnet to which an armature is fitted. If the surfaces of the pole, and armature be carefully ground flat, so as to fit, we may regard the magnet and the armature as continuations of each other. The force of cohesion here is mainly due to the magnetism; and the force required to separate the two is called the “carrying power” of the magnet. To simplify the question, let us consider a cylindrical bar of section ω, uniformly magnetized in the direction of its length with intensity I. Suppose the bar cut so that the normal to the plane of section makes an angle θ with I, and let the surfaces of section be separated infinitely little, then the surface density of Poisson’s distribution will be I cos# on each surface. Assuming that the cohesion is caused solely by the attraction of these surface layers, we get for the carrying power P = 2τrlcos0 × Icos0ωsec0, 7.e., P = 2τrl 2 ωcos0. The carrying power is therefore greatest, viz., 2τrωl 2 , when the surface of the pole is perpendicular to the lines of magnetization.

A great variety of experiments have been made on this subject by Joule, Dub, Tyndall, Lamont, and others, mostly, however, under circumstances that do not admit of the application of the above theory. For an account of what lias been done, the reader should consult Wiedemann’s Galvanismus, ii. § 425 sq. The most recent investigations on the subject will be found in the papers of Rowland, quoted below, p. 255, and in papers by Stefan and Wassmuth in the Monatsberichte der Weiner Akademie for 1880 and 1882. 2 The facts are not so simple as the above theory would indicate; but Wassmuth finds a modified form of it to agree sufficiently well with observation.

Mathematical Theory of Magnetic Induction.

The two fundamental axioms of this theory are the following :—

1. The induced magnetism in any element of a body depends merely on the magnitude and direction of the resultant magnetic force (∣)) at the element.

2. The magnetic moment induced by any force ∣j is the resultant of the magnetic moments induced separately by any forces of which ∣∣ is the resultant.

With reference to axiom 1 it is to be remarked that account must be taken of the physical condition of the body as to temperature, and so forth; but it is implied that no account is to be taken of its magnetic state, except in so far as that affects the resultant magnetic force. In other words, it is asserted that the moment induced by any force does not depend upon any pre-existing magnetic moment in the element, and is the same whatever forces may have acted on the element previously. The full significance of these statements will be better appreciated when we come to consider the exceptions to them in case of strongly magnetic bodies. It should also be noticed that it is supposed that the body has reached a state of magnetic equilibrium, and that by whole resultant magnetic force is understood, not only that arising from the given inducing system, including pre-existing magnetism in the body itself, but also that arising from induced magnetism.

In the mathematical theory no distinction is drawn between the part of the induced magnetism which disappears when the inducing force is removed, and that which remains. If anywhere we contemplate what happens after the removal of the force, it is assumed that all the induced magnetism disappears. This important restriction must be borne in mind in applying the results in practice.

Axiom 2 enables us to assign at once the law connecting the components of induced magnetization A 1 , B 1 , C 1 with the components α, β, γ of the resultant force. If r 1 , ç 3 , p 2 be the components parallel to the three coordinate axes of the induced magnetization caused by a unit resultant force parallel to the axis of x, then, by the axiom, the components of magnetization induced by a force α in the same direction will be r 1 α, <7 3 a, ρ 2 α; similarly, if ρ 3 , r 2 , q 1 be the components due to unit force parallel to the y axis, then the components due to β will be ρ 3 β, r 2 β, q l β; and -finally, if q 2 , p l , r 3 be components due to unit force parallel to z axis, the components due to γ will be q 2 γ, p 1 γ, r 3 γ. Compounding all the⅛e, according to the axiom, we get

A 1 = r 1 α+^ 3 3 + g 2 γ }

B 1 = g s α + r 2 j8+^ 1 γ > (71).

C 1 =p 2 a + g- 1 0 + r 3 γ )

Hence the most general expressions for the components of magnetization compatible with our axioms are three linear functions of the components of the resultant force.

Here it is necessary to introduce a classification of bodies according to their magnetic properties.

If equal, similar, and similarly situated elements cut from different parts of a body have identical magnetic properties, it is said to be “magnetically homogeneous,” if not, “heterogeneous.”

If equal and similar elements cut around the same point in different directions be identical in their magnetic properties, the body is said to be magnetically “isotropic”; if not, “aeolotropic.”

These are not cross classifications; for a body (e.g., Iceland spar) may be aeolotropic and yet homogeneous, and it might be heterogeneous and yet isotropic. We must regard the coefficients p, q, r of (71) as belonging to a point of the body; and we see that in a homogeneous body they will be the same for all points, whereas in a heterogeneous body they will vary from point to point, i.e., they will be functions continuous or discontinuous of the position of the point.

In the case of an isotropic body it is obvious a priori that the induced magnetization must be coincident in direction with the resultant force; the conditions for this are that the coefficients p and q should all vanish, and that r 1 =≡.r 2 = r 3 = κ. The equations (71) thus reduce to

A 1 = κα, B 1 = κ∕3, C 1 =κγ .... (72).

In an aeolotropic body, on the other hand, the coefficients may be all different from zero and from one another; but, as we shall see, at all events in the ideal case contemplated by the mathematical theory, the conservation of energy reduces the number of independent constants by three; while a proper choice of axes reduces it by three more; so that the magnetic properties of any element of an aeolotropic body depend virtually on three independent constants.

The theory here given is the generalization of Poisson’s theory due to Sir William Thomson. It aims at giving the simplest possible exposition of the results of experiment with the fewest assumptions as to the molecular structure of bodies. We first discuss specially a few of the cases 1 Various experimenters have attempted to determine the “indifferencezone” of magnets under different circumstances, i.e., the line separating the positive and negative parts of the surface distribution. For information as to this and other matters under the present head omitted for want of space, see Wiedemann, Galvanismus, ii. §§ 277, 356, 396, 401; and Lamont, Handbuch, §§ 6, 27, 63, 64, 65.

2 Abstracted in Wied. Beibl. 1880 and 1882; see also Von Waltenhofen, Wien. Ber., 1870, and Siemens, Bert. Mοnatsber., 1881.

more important in practice, and then give a brief account of the general theory with a view to establish some general principles to guide us in the subsequent account of the (often very complex) phenomena observed by experimenters.

Homogeneous Aeolotropic Sphere in a Uniform Field of Inductive Force.— We suppose that the sphere, to begin with, is not magnetized. If the sphere were uniformly magnetized, with components A 1 , B 1 , C 1 , then (see above, p. 232) the force inside the sphere due to this magnetization would have for its components

βι= - ⅜ π A 1 , 3 1 =-⅜tB 1 , 7 1 =-⅜πC 1 .

This uniform force combined with the given uniform force (α 0 , ∕2 0 , γ 0 ) of the inductive field would result in a uniform force

α = ⅝-⅛'rA 1 , 3 = 0o~⅜ ιr Bι> 7=7o - ⅜ 1 rC 1 ι · · (73).

It is obvious therefore that the assumption of uniform magnetization will enable us to satisfy the law of induction.

In point of fact, substituting in (71) and transposing, we get three linear equations to determine A 1 , B 1 , C 1 in terms of α 0 , β 0 , 7 c , viz.,

(1 + ⅜∙π∙r 1 )A] +⅜πp 3 B] + ⅜ 7γ⅛ , 2 C 1 — r 1 a o +p30 o + g , 2 7 o , &c. . (74).

It is easy, by means of these and formulae given above, to calculate the couple exerted on the inductively magnetized sphere. If we put α o = 0, ∕3 o = Fcos0, γ 0 = FsinØ, we can calculate the work done on the sphere in turning through 180° about an ax⅛ perpendicular to the direction of the field. This, by the conservation of energy, ought to vanish, and we thus get the conditions p x = q x , ρ 2 = 9 2 , p 3 = 7 3 . The equations (74) therefore reduce to

A 1 =r 1 α+i⅛3+ j p 2 γ ∖

B 1 =p 3 a + r 2 3 + p 1 7 ’ (75).

Cι=^ 2 a+^ι0 + r 3 7 )

∏ence, if a, β, y be parallel to a radius of the central quadric r 1 aι 2 + r 2 y 2 4- r 3 z 2 + 2p 1 yz + 2p,px + 2p 3 xy =1,

A 1 , B 1 , C 1 will be normal to the diametral plane of that radius. We have, therefore, by the theory of surfaces of the second degree, the following conclusions.

1. The induced magnetization is not in general in the direction of the inducing force; but there are in general at every point three directions, called the three principal magnetic axes, mutually at right angles to each other, for which the directions of the induced magnetization and of the inducing force coincide. If the axes of coordinates be parallel to these principal axes, the equations (75) reduce to

A j = r 1 α, B 1 =r 2 3, C 1 =γ 3 7 .... (76).

The values of r 1 ,r 2 , r 3 in this case are called the “principal magnetic inductive susceptibilities.” Bodies for which these coefficients are all positive are called paramagnetic or ferromagnetic. Bodies for which they are all negative are called diamagnetic. No substance is known for which some are positive and others negative, although this is a mathematically possible case. Since intensity of magnetization and resultant magnetic force are of the same dimension βL *M i T 1 J, r 1 , r 2 , r 3 are pure numbers; for all substances except iron, nickel, and cobalt, they are extremely small. The value of the coefficients r and p for any other axes can be expressed in terms of the three principal susceptibilities by means of simple formulae which we need not stop to deduce.

A physical meaning can be given to r 1 , as follows. Let the body be homogeneous, and let us cut from it a cylindrical piece whose axis is parallel to the principal axis of susceptibility r 1 . Place this cylinder in the direction of the lines of force in a uniform field of unit strength, then, provided the cylinder be infinitely thin, and of longitudinal dimensions infinitely great compared with its lateral, the internal force due to the induced magnetization will be zero (see above, p. 229), and it will be magnetized inductively with a uniform intensity r 1 . Similarly for r 2 , r 3 .

The three coefficients w 1 = l+4πr 1 , α ’ 2 = 1 ÷ 4τrr 2 , w 3 = l÷4πr 3 , used later on, are called by Thomson the three principal permeabilities of the body at any point. These are of course pure numbers, and they are positive for all known substances.

2. If the susceptibilities for any two principal asis be equal, then every axis in the plane of these two is a principal axis.

3. If all three principal susceptibilities be equal at any point, then every axis through that point is a principal axis, and the susceptibility for every such axis is the same. The body is therefore isotropic at that point, and the direction of the induced magnetization coincides with the direction of the inductive force for every direction of the latter.

Returning to the problem of the aeolotropic sphere, let us simplify our equations by taking the coordinate axes parallel to the common directions of the principal axes throughout the homogeneous sphere. We then get for the components of magnetization a i=Γ⅛→o> c ι~VΓ⅛'*> · (77) · jl + ^7γγ 1 ι+ π τrr 2 i + ιj∙tγγ 3

Using these formulae, we get, by means of (22), for the components of the couple acting on the sphere (of volume υ),

^∖ι+⅜⅛l⅛^ 0

^ = r (l+√ 3 )(l+⅜πr 1 ) 7 θ α ° · · (78) ·

J

There is of course no force of translation. As a special case let us suppose- r v r 2 , and r 3 to be in descending order of algebraical magnitude, and suspend the sphere with the axis of r 1 perpendicular to the lines of force. We may put β o = Fcos0, γ o = Fsin0, where θ is the angle between the axis of γ (r 2 ) and the direction of the field, then we have ί = ⅜√F 2 (r 2 - r 3 ) sin 20∕(1 + ⅜wr 2 )(l + ⅜wr 3 ).

Hence the sphere tends to turn so as to place the axis of algebraically greatest susceptibility parallel to the lines of force. It will be in equilibrium when either principal axis is parallel to the lines of force; but in stable equilibrium only when the axis of greatest permeability is in that position. It is to be noticed that the couple is proportional to the square of the strength of the field.

There is another way of expressing these results more in accordance with the ideas of Faraday.

If N be the surface integral of magnetic induction taken over the meridian section («) of the sphere perpendicular to the direction of the vector inside, or, as we may call it, the number of lines of force that pass through the sphere, then we have

N = 3Fω J (gf2 ⅛) ’ cos 2 0 + (—⅞s) ≡in 2 0 I

( ∖w 2 + 2∕ ∖w 3 + 2∕ 1

¾(=- 1 (<r t +2)(tr 8 +¾) d(N 2 ) j

24jΓ⅛ ∙≡r 2 + ≡r 3 + ∙≡r 2 τV3 Clθ ’

R being the radius of the sphere.

From these formulae we can draw the following conclusions :—

1. The number of lines of force that pass through the sphere is greatest, viz., 3Fωτr 2 ∕(π 2 + 2), when the axis of greatest permeability is parallel to the direction of the undisturbed field, and least, viz., 3Fωw 3 ∕(w 3 +2), when the axis of least permeability is in the same position.

2. In any position the number of limes passing through the spherical space is greater for a paramagnetic body, and less for a diamagnetic body, than it would be if the sphere were absent.

3. The sphere is in equilibrium when the number of lines of force passing through it is a maximum or a minimum, the equilibrium being stable in the former case, and unstable in the latter. Homogeneous Isotropic Sphere in Uniform Field.— This case is obtained by putting r 1 = r 2 = r 3 = κ in the above formulae. The magnetization is parallel to the undisturbed field; and the couple vanishes, so that the sphere is in equilibrium in all positions. If the strength of the field be F, we get for the intensity of magnetization

I = _ f = s ( 9 ~ 1 ⅛;

1 + - 4 ™ 4 w+2 ’

also xr 3τr r ~ l

M =; ι ’ ω. τsr+ 2

In order to familiarize the reader with this important case, we give two figures of the lines of force from Sir W. Thomson’s Reprint, pp. 490, 491,—one for a paramagnetic.

(fig. 32) having w = 2 ’ 8, and another for a diamagnetic (fig. 33) having ≡r = ∙48. The former represents a paramagnetic whose susceptibility is something like τ ∣ 17 th of the maximum observed for the best Norway iron. The latter corresponds to a diamagnetic having a susceptibility some 16,000 times that of bismuth, which is the most powerfully diamagnetic substance known.

The reader should observe that, although the field inside the isotropic sphere is uniform, this is not the case outside, a fact sometimes forgotten by experimenters. Of course the disturbance in the case of a bismuth sphere would be infinitesimal.

Homogeneous Aeolotropic Ellipsoid in a Uniform Field. —In the case of a sphere the tendency to set in a uniform field is wholly dependent on the aeolotropy of the sphere, and is independent of its form. It is important, in order to get a complete picture of the behaviour of inductively magnetized bodies, to obtain a solution for some case where the form has an effect upon the result. A solid bounded by a surface of the second degree affords such a case.

If an ellipsoid be uniformly magnetized so that the components of magnetization parallel to its three principal axes α,b,c be A 1 ,B 1 ,C 1 , this magnetization gives rise to a force

α 1 =-A 1 I., 0 1 =-B j M, γ 1 =-C l Nj when I.. M, N have the values given above, p. 232. If we now place this ellipsoid in a uniform field (α 0 , β 0 , γ 0 ), the force inside will be given by a = a 0 -A 1 I., 3 = 3 0 -B 1 I., γ =γ 0 -C 1 I. . . (79).

It is obvious, therefore, that the equations (75) of induction can, as in the case of a sphere, be satisfied by the assumption of uniform magnetization.

There is no difficulty in dealing with the general case in which the principal magnetic axes are not parallel to the principal axes of figure; we shall content ourselves, however, with the case in which the principal magnetic axes r 1 , r 2 , r 3 are parallel respectively to a, b, c. Equations (76) then give at once

A 1 =r 1 (α 0 -A 1 I.), B 1 = r,jβ 0 - B 1 M), C 1 ≡r 3 (γ 0 -C 1 N) (80); whence a i = ΓT‰ b i = t z ⅛> C 1 - τ ¾ r . . (81).

l + r 1 I. 1 l + r 2 M ’ 1 l + r s N

The components of the magnetic moment are of course obtained at once from these by multiplying by the volume.

For the components of couple, ^t, gl, tending to turn the ellipsoid about the axes a, b, c, we get

⅝> 4 π -.~r- r ¾ ~ r 3 + r 2 r 3(N - M))

J 7r παδc + 3 0 γ 0 ∕ 1 . . (g2)

$J = &c., $ = &c. 1

From these equations we can draw the following important conclusions,—first as to the magnetization of the ellipsoid.

1. When r 1 , r 2 , r 3 are so small that their squares may be neglected, as in fact is the case with all bodies except iron, nickel, and cobalt, the components of magnetization reduce to r 1 α 0 , r 2 β 0 , r 3 γ 0 . A glance at equations (79) will show that what happens is simply that the part of the internal inducing force which depends on the squares of the susceptibilities is not sensible. In other words, the form of the body is without influence on the induced magnetization. Or, what is again equivalent to the same thing, the induced magnetism may be supposed to produce no disturbance in the inducing field.

These conclusions are of course not limited to the ellipsoidal form in particular; but we have the general result that, if the squares of the susceptibilities are negligible, then the form of the body has no effect on the induced magnetism.

2. On the other hand, when the susceptibilities (and consequently the permeabilities) are very great, since A 1 = α 0 ∕(1∕r 1 + I.), &c., it is clear that the influence of the form of the body predominates. The extreme case is that of a body of infinite permeability, in which the induced magnetism is wholly determined by the form.

3. If, however, the ellipsoid be very elongated in the direction of a, then I. will be very small, and r 1 I. may be very small, notwithstanding the largeness of r 1 . In that case A 1 = r 1 α 0 .

4. From 1, 2, and 3 we have the following most important results. In experimenting with weakly magnetic bodies in a uniform field—in order, say, to determine their susceptibility—the form of the body is indifferent. On the other hand, with strongly magnetic bodies an elongated form must be used, because in that case only does the induced magnetism depend mainly on the susceptibility oi the material. With bodies approaching the spherical form differences in form produce far more effect on the experimental results than differences in the susceptibility of the material, so that in such cases the experimenter really measures the accuracy of his instrument maker more than the magnetic susceptibility of his material.

5. For a flat disk (infinitely oblate ellipsoid), having its r 1 axis parallel to the lines of force, I. = 4τr, and A 1 = r 1 αθ∕(l + 4τrr 1 ) = a 0 (τv 1 - 1)∕4τr∏r 1 . If such a body were diamagnetic, and had r 1 = - 1∕4τr, t.e., had zero permeability, the normal magnetization would be infinite for any finite force.

Next, we have the following conclusions as to the magnetic couple. Let us suppose the ellipsoid free to move about its a axis, and let the direction of the field be perpendicular to a, so that ∕3 o = Fcos0, γ o = Fsin0j the couple tending to turn the δ axis parallel to the undisturbed direction of the field is the sum of two parts :—

¾-¾√ α ⅞--U ≡2β · · < 83 >' 34 3 (1+ r a M)(l+ r 3 N)

and

πα⅛r a r 3 (N-M) .... (84).

s (l+r 2 M)(l+r 3 N) 8inz0

1. If the susceptibilities are so small that their squares and products are negligible, then reduces to

9f 1 =⅜παδcF 2 (r a - r 3 ) sin 20.

,In other words, the form of the body has no effect, and it behaves exactly like an aeolotropic sphere of the same volume; i.e., it will tend to turn its axis of greatest permeability parallel to the lines of force.

2. If the susceptibilities be very large, then the most important part of will be ∣Γ 2 . Now a glance at the values of M and N (34) shows that N — M has the same sign as 6 2 — c 2 ; hence the ellipsoid will tend to place its longest dimension parallel to the lines of force d This is the general effect of the influence of form in the case of strongly magnetic bodies, or, if we choose to put it so, the effect of the disturbance of the field by the induced magnetism.

3. It is of course possible in the case of strongly magnetic bodies that both parts of may be sensible, so that the resultant action would be affected both by form and by the magnetic structure, either predominating according to circumstances; for by properly shaping the ellipsoid we can give N - M any value positive or negative from 0 to 2τr. In this way, given an aeolotropic body for which 1∕r 3 - 1∕r 2 is not greater than 2τr, we might so shape it that it would turn its longest dimensions parallel to the lines of force, or so that it would turn its shortest dimensions parallel to the lines of force, the shortest axis in the second case being the axis of greatest permeability; or we might so shape it that the equilibrium would be neutral.

And, in general, given a body aeolotropic within certain limits, we might shape it in such a manner that the effect of its form would exactly neutralize the effect due to its structure, so that, as far as setting in a uniform field is concerned, it would behave like an isotropic sphere.

Homogeneous Isotropic Ellipsoid in a Uniform Field.— The formulae for this case are of course at once obtained by putting r 1 = r 2 = r 3 = κ in the above formulae. We thus get

A - καn B - κ3 ° C ≈ κy °. · · (85): ai ~1 + kL , bl 1 + kM’ 1 l + κN k h gt = &c., $ = &c. (86). (1 + kM)(1 + kN)

The following conclusions are worthy of notice :—

1. The resultant magnetization will not, as in the case of an isotropic sphere, be parallel to the resultant magnetic force.

2. The ellipsoid will tend to set its longest dimension parallel to the lines of force; and, since κ is involved in the numerator in the form κ 2 , this conclusion is the same for a diamagnetic as for a paramagnetic body. Of course we exclude the mathematically possible case of one of the factors 1 +κM or 1 + kN vanishing or becoming negative. For all weakly magnetic bodies, however, κ is so small that the tendency of an elongated isotropic body to set in a uniform field is insensible.

Ping Electromagnet.— A simple case, which has recently acquired practical importance, is that of an electromagnet having a soft iron core, shaped like an àncbor ring, whose mean diameter is R, and radius of section a, wound uniformly with n turns of a primary coil in which flows a current z'. The lines of force and the lines of magnetization will evidently be circles, and, since the Poisson’s surface and volume distributions vanish, the whole magnetic force ∣) will be simply that due to the current. At a distance ρ from the axis of the ring ∣J = 2m∕p; for the whole work done on a unit pole in passing round any coaxial circle of radius p is $x 2τrp = 4τrτw'. The intensity of magnetization is, therefore, I = 2nκi∕p, and ∣∣ = 2ni(4ττκ+ l)∕p = 2n^i∕p. Hence it appears that the total induction through a secondary coil of n' windings is 2ηn'i(4πκfdS ∣ p+fdS' ∣ p), where fd^>jp is taken over the section of the core, and fdS' ∣ p over the section of the coil. In the case of an anchor ring of circular section, if we neglect the difference between the radius of the primary coil and the radius of the core, the expression for the total induction through the secondary is 4ιmn z m(R - x ∕R 2 — α 2 ).

In a non-uniform field the problem of magnetic induction becomes very difficult for bodies of finite size. If, however, we deal with infinitely small bodies we may suppose the field uniform throughout the body, and apply the results already obtained to find the induced magnetism.

Small Aeolotropic Sphere in a Non-uniform Field— Let A 1 , B 1 , C 1 be the components of the induced magnetization parallel to the principal magnetic axes of the sphere, α o ,0 o >7o the components of the strength of the undisturbed field at the centre of the sphere in the same direction; then, denoting r 1 ∕(l + ⅜7rr 1 ) by s 1 , and so on, we have A 1 =s 1 α 0 , B 1 =s a 0 o , C 1 =s 3 γ 0 . If the magnetization of the small sphere (of volumes) were rigid, its potential energy w would be w= -t>(A 1 α 0 + B 1 ∕3 ft +C 1 γ 0 ). The actual potential energy, W, of the inductively magnetized sphere is different, because its magnetism varies as it passes from one part of the field to another. In any infinitely small displacement, however, we may calculate the work on the supposition that the magnetism is temporarily rigid. In other words, we may put <ZW = rfW', where the latter is taken on the supposition that A 1 , B 1 , C 1 do not vary, while on the other hand a 0 ,β 0 ,γ 0 do vary, because the resultant force both alters its magnitude and its direction relative to the principal axes of the sphere. We thus get

(ZW = - i)(A 1 dα 0 + B 1 dβ 0 + C 1 dγ 0 ).

In integrating we must take account of the fact that A 1 ,B 1 ,C 1 are variable. Substituting their values, we get d W = - 'r(s 1 α 0 dα 0 + s i β 0 dβ 0 + S 3 7 0 dγ 0 ), whence W=--‰o 2 +Mo 2 + ⅞7o s ) · · · (87).

Δ

This important formula contains the whole of the theory of the movement of small spherical masses of inductively magnetizable matter in any field of force. We can deduce from it at once the position of equilibrium of an aeolotropic sphere suspended in a uniform magnetic field, with freedom to rotate about a given diameter.

Let λ, j u, V and l, m, n be the direction cosines of the given diameter and of the direction of the field relative to the principal magnetic axes of the sphere, and R the strength of the field; then W = -⅜ι>R 2 (s 1 Z 2 + s a wι 2 + s 3 n 2 ). For stable equilibrium W must be a minimum, and for unstable equilibrium a maximum, i.e., there is stable or unstable equilibrium according as s 1 Z 2 + s. 2 m 2 + s 3 n a . is a maximum or a minimum under the given kinematical conditions, which will be expressed by a relation between λ, μ, ν and l, m, n. It is needless to work out the analytical solution; for it leads to results easily obtainable from formulae already given. It is important, however, to show the identity of this method of treatment with Faraday’s view of the matter. If a, b, c be the components of the magnetic induction parallel to the principal axes of the sphere, then we get α=(l+⅞π∙s 1 )R7, 5 = &c., c=&c. Whence if N denote the total induction through the sphere 1 in the direction of the undisturbed field, we have, ω being the area of its meridian section,

N = ω(yl + bm + cn)

= R + f ττR(s 1 Z 2 + ⅞wι 2 +⅝n 9 ).

N is thus a maximum or a minimum when s 1 l i + s a τn 2 + ¾π 2 is a maximum or a minimum.

We have therefore established quite generally Faraday’s law that an aeolotropic sphere suspended in a uniform field with freedom to rotate about any diameter will be in stable or unstable equilibrium according as the number of lines of force that pass through it is a maximum or a minimum. A particular case of this theorem has already been proved above (p. 245) for strongly magnetic bodies.

We next apply the formula (87) to deduce the force of translation in a heterogeneous field.

1. We see that in a uniform field W is constant so long as there is translation merely without rotation, i.e., there is no tendency in an aeolotropic or isotropic sphere to move bodily in a uniform field.

2. If we suppose the sphere isotropic (Le., s 1 = s 2 = s 3 = τ), then W = - ∣τv(αθ +βο+ γ 2 ) = - ⅜τ√R 2 . Hence the force tending to move the sphere in the direction of ds is j clW . d(R 2 ) _dR , ss s ds i ds ds

In other words, the small sphere is subject to a force of which the scalar potential is ∣τvR 2 . If then we draw the isodynamic surfaces R 2 = const., the force on an inductively magnetized isotropic sphere will be everywhere at right angles to these; that is, the direction of this force at every point will be tangential to the lines of slope of the resultant force, viz., in the direction in which that force varies most rapidly. In the case of paramagnetic bodies, for which τ is positive, the spheres will tend to move from places of weaker to places of stronger resultant force ; in the case of diamagnetic bodies, for which τ is negative, from places of stronger to places of weaker force. This is the famous law found experimentally by Faraday, and afterwards theoretically established by Sir William Thomson.

It must be carefully borne in mind that the lines of slope of the field are not necessarily coincident with the lines of force, but may cross them at any degree of obliquity.

As strange mistakes have been made in this matter, 2 it may be well to illustrate this statement by a few examples.

In the case of an isolated north pole the lines of slope coincide with the lines of force, which are straight lines radiating from the pole. In this case a paramagnetic sphere would approach and a diamagnetic sphere recede from the pole along the lines of force.

The lines of force for a rectilinear electric current are circles of which it is the axis; the lines of slope are straight lines radiating from the current. A paramagnetic sphere would therefore move towards, a diamagnetic sphere away from the current in a direction perpendicular to the lines offorce:

In the case of an infinitely small magnet, whose lines of force are given by r=c sin 2 0, the lines of slope are given by r=c' sin 4 0∕ cos 0; and the angle between the line of force and the line of slope at the point (r,0) is tan ^ 1 {tan0(l + cos 2 0)∕(3 + 5 cos 2 0∕}

The theory of isodynamics and lines of slope in the case of plane fields of force, i.e., those for which the potential is given by the equation d 2 V∣ dx* + d 2 N ∣ dfi=0, is remarkably simple. If ξ=x+zy, ιη=x-iy, we know that V = ≠(ξ) + ψ(η), φ and ψ being functions depending on the particular case. When these are known the isodynamics are given by 1 , 1 . , Φ'(ξ)Ψ'(η) = const.

and the lines of slope by ffl-eonst.· ψh)

3. Next suppose an aeolotropic sphere allowed to move without rotation in any direction ds. Let the direction cosines of the field relative to its principal magnetic axes be l,m,n, then these are constant during the displacement; and, if R be the intensity of the field, α 0 = R∕, β 0 = R7n, γ 0 = Rraj whence

-^→(√ a +^n 2 + ⅞n 2 )^ · · · (89).

Hence, as before, the resultant force of translation cm the sphere is along the line of slope, in the direction in which the force increases if the body be wholly paramagnetic, in the opposite direction if it be wholly diamagnetic.

Besides depending on the nature of the field, the force of translation, on account of the factor sf 2 4- s 2 m 2 + s 3 n 2 , depends on the position of the body relative to the lines of force. Bearing in mind the theory of the radii of an ellipsoid, we have the following proposition :—

The force of translation on an aeolotropic sphere is greatest when its axis of (numerically) greatest magnetic susceptibility is parallel to the lines of force, and least when the axis of (numerically) least susceptibility is in the same position.

Or, using permeability instead of susceptibility,—

The force of translation is greatest for a paramagnetic sphere when its axis of greatest permeability is parallel to the lines of force, for a diamagnetic sphere when the axis of least permeability is parallel to the lines of force, and vice versa.

Or, yet again, in the words of Faraday :—

The force of translation exerted upon a paramagnetic sphere is greatest when it is so placed that the greatest number of lines of force pass through it, whereas in the case of a diamagnetic sphere the force is greatest when it is so placed that the least number of lines of force pass through it, and vice versa.

Approximate Theory of the Action on Bodies of Finite Size in a Non-Uniform Field.— We have seen that, if the square of the susceptibility be negligible, the effect of the form of the body and the disturbance of the field arising from the induced magnetism may be neglected. In that case we may replace the spheres of the foregoing discussion by cubes, and determine the action on a body of finite size by integrating the action on the elementary cubes of which it is composed. Thus the potential energy will be - ⅛fff(s 1 αQ + s 2 β 2 + s 3 yl)dv, and the body need not necessarily be homogeneous. From this expression we can deduce the force under given circumstances.

It is quite easy to see, without any mathematical calculation, what will happen in a field of force which diminishes in intensity outwards from an axial line. If we suspend an elongated paramagnetic body with its centre in the axis of the field, it will evidently be in stable equilibrium with its longest dimension placed axially; for if it were slightly displaced every little cube of it would move into a place of weaker force, and would therefore tend to return. If, on the other hand, the body were diamagnetic, it would be in stable equilibrium in an equatorial position; for any displacement from that position would bring every little cube nearer the axis of the field, i.e., into a place of stronger force, and therefore each such cube would tend to return.

General Problem of Magnetic Induction.— It will be instructive to consider for a little the theory of induced magnetism in its most general form.

We shall suppose the induction to arise from given magnetic force (α o ,0 o ,γ o ), arising from pre-existing magnetism (A 0 ,B 0 ,C 0 ) or otherwise. Letters with suffix 1 denote components of induced magnetism, of force arising therefrom, and so on. Letters without suffix denote components of total force, total magnetization, &c. Thus V o , V 1 , V denote the potentials due to pre-existent, induced, and total magnetism respectively; and we have V = V 0 + V1, and the like relation in other cases.

We suppose all the media within the field to have definite permeability; but there may be aeolotropy and heterogeneity to any extent, and discontinuity along given surfaces.

Resolving along the principal magnetic axes at {x, y, z), we get, by the law of induced magnetism, (l l , m v n 1 ), ‰ m i , n 2 ), (¾, m i , n 3 ) being the direction cosines of the axes w 1 , ∙sr 2 , τ≡∙ 3 , al l + bm 1 + cn 1 =βr 1 (αZ 1 + βm 1 + γn 1 y ) + 4π(A 0 Z 1 + B 0 wι 1 + C 0 n 1 ), and two similar equations.

Multiplying these by l 1 , l«, l 3 , adding and so on, we get [table]

where s ι ~ zr xlι 2 ÷ ^ a fι + " a f∙f > ~ w 1 wι 1 τι 1 + w 2 nι 2 ra 2 + w 3 nι 3 π 3 ,

&c.;—that is to say, given functions of x,y,z.

Besides these, we have the conditions of normal continuity for $, viz., da i db j de θ dx ^ r dy ^ r dz , and at a surface of discontinuity, λ, μ, v being the direction cosines of the normal at any point from the first medium to the second,

(a - a')λ + (δ- b')μ + (c -c')v=0, dashed letters referring to values on the first side of the surface, undashed letters to values on the second.

From these we get finally, for the determination of V.

Jr-(*173r + t a ^ + < a ^) + &c. +&c + 4τrp o = O . (91). dx ∖ 1 dx 3 dy 1 dz ) and ( ,dN' dV ι t ,dV' t dV ι . ,d ∖ τ ' . dV)

1 5ι Z Sι j *τ" Γq T” “· vo . ~Γ r J z j ∖ A

( 1 dx 1 dx 3 dy 3 dy dz i dz )

+ &c. μ+ ∣ &C. >j'+4π<r o = O . . (92).

Here

_ ∕ ’ dAθ ι dB 0 ι tZC⅜ ∖ p °~ ∖ dx + dy dz J ’ <r 0 ~ (A® - L0 ΙΑ ^t (Bø ~ B® )∕*^t (θθ — 5

‰e., they are Poisson’s volume and surface densities for the preexisting magnetization.

It may be shown by a method essentially the same as that used in the article Electricity, vol. viii. p. 27, that equations (91) and (92), with the condition that V be continuous everywhere and vanish at infinity, lead to a unique determination of V. When V is known, A 1 ,B 1 ,C 1 can be found at once from (90).

In what follows we shall confine ourselves to homogeneous isotropic media, and we shall suppose that in parts of the medium inductively magnetizable there is no pre-existing magnetism.,. The equations (91) and (92) then reduce to

(93 ), αx i dy dz i and

<ZV , ' rfv ' a can a - r - + w -z- r = 0 (94), dv dv dv,dv' being elements of normals drawn inwards in the two media. Equations (90) reduce to a = -era , b ≈ τrβ , C = wγ; whence

_.-1 dV b _η 2 ι rfV .-ι riV 4τr dx 4n∙ dy 4τr dy

From these last, combined with( 93), we have the important consequence that the induced magnetization is both solenoidal and lamellar. This is true only for homogeneous isotropic media in which the pre-existing magnetism, if any there be, is solenoidal.

In all cases such as we are now considering, the part of the magnetic potential due to induced magnetism may be calculated wholly from surface distributions at the surfaces of discontinuity. If σ 1 be this surface density, we have ^+≤^ + 4τ<r 1 = 0 (96).

dv dv 1

From equations (94) and (96)

dN ,dN' σ ^ ,c >dv= κι dff> where ιr i = (w - zr h ) ∣ 4ιrw', κ 1 ' = (w - tv) ∕ 4τrw.

Let us suppose that a body A of permeability -ar is sus pended in a medium of permeability -≈'. If the susceptibilities be small,' the forces arising from the induced magnetism will be so small that the direction of the normal force at the surface of A will be the same as if the field were undisturbed by its presence.

First, let the medium be vacuum, for which we suppose √≡1, then, if A be paramagnetic (i.e., w>1), κ 1 will be positive, and the surface magnetism will be positive where the lines of force leave the body, and negative where they enter it. If A be diamagnetic (i.e., w<l), κ 1 will be negative, and the magnetic polarity of the body as a whole will be opposite to what it was in the former case.

Secondly, let the surrounding medium have permeability w, then κ 1 is positive or negative according as w > or <ιsr'; in the former case A will behave like a paramagnetic body in vacuo, in the latter like a diamagnetic body in vacuo.

It appears then that, by virtue of differential action, a body may behave paramagnetically or diamagnetically according as it is placed in a less or in α more permeable medium than itself

In practice it is most convenient in general to determine V 1 instead of V. The above equations can be easily modified to admit of this. In fact we get at once, remembering that dNj∕dv'≈ -dNfdv, since V® has no discontinuity at the surface of the media,

⅛ + <*v <τw 1 (97); dx i + dy* + dz* ' 7 ’ dTl , > dVj. j >∖ f W 0 ∕∙qo∖

"* +w *Γ + <"-^>⅛-°∙ · · · !98) ·

These equations, together with the condition that V 1 be finite and continuous and vanish at infinity, determine V 1 completely. Since the induced magnetization is lamellar, we may write k 1 =dφ 1 ∣ dx &c., we then get by (95)

≠ 1 =-⅛(V 0 + V 1 )=-∕ t (V 0 + V 1 ) · · . (99), u7Γ

which give the components of moment in terms of the known function V.

The number of cases in which the solution of the induction problem can be worked out is very small. Besides those already treated synthetically, one or two more, affording examples of the general method, will be mentioned in the historical summary below. Meantime, we must not omit to mention an extremely elegant transformation theorem, due to J. Neumann, which enables us to deduce the magnetic moment of any body as a whole under the action of any forces whatever, when its magnetization in a uniform field is known.

Let A 1 ',B 1 ',C 1 ' be the components of induced magnetization produced in the body A by the uniform field whose components are α 0 ',3 0 ’,<y 0 ’! A 1 ,B 1 ,C 1 the magnetization produced in A by any field whatever (α 0 ,⅛ 0 ,γ 0 ). Let fdυ denote volume integration throughout A; and consider the function

IT =Jdv{α a k^ + 3 0 B∕ + 7oθ∕) ·

IT is known, since we suppose α 0 ,j8 0 ,-y 0 , A 1 ',B 1 ',C 1 ' to be known. But, since A 1 = κ(a 0 + a 1 ), &c., A∕ = κ(a 0 '+ a 1 '), &c., we have V =J~dv I <⅛^κ(a 0 '+ a 1 ') + &e. · · · |

=fdv{ajk 1 + &c. ) + fdυlajk 1 + &c. ) -Jclv{a l k{ + &c. ).

Now the last two terms destroy each other; for they are simply different expressions for the mutual potential energies of the induced magnetism due to (α 0 , B o , γ 0 ) an d to (α o , >. 3o'> 7o')> regarded as separate rigid systems, although coincident in position. Hence we 5 et

IT= a 0 'f^dvk 1 + β' a fdυ⅛ 1 + y 0 fdvC 1 , whence ∕dυλ,≈- r -,, = ∕<irC 1 = v-η⅛.

√ l da 0 ' , J dβ 0 J dγ 0 For an ellipsoid this gives at once for the components of moment p -Γ⅛c∕' ’ ' α ∙' κ -Γ⅛N√ 5cγ ∙∙

An interesting particular case is that of an infinite cylinder. If P be the component of the moment parallel to its axis, P= - κffdydz (V β0 -V- 00 ). If the inducing system be magnetic bodies at a finite distance, then V 0o = V- β0 = 0, and P = 0. If the cylinder be magnetized by a spiral current i, of n windings, of whatever form, then, r being the radius of the cylinder, P = 4τr 2 κz⅜i Generalization of the Theory for Isotropic Media in which κ is not constant.— In the above theory we have supposed the magnetic susceptibility to be constant. This is by no means the case in nature, however. It is of importance therefore to consider how the theory must be modified when we assume κ to be a function of the magnetization already induced. Subject to the restriction (obviously necessary for isotropic media) that the resultant magnetization shall coincide in direction with the total resultant magnetic force ('∣J), the most general assumption that can be made is

A-∕(I)α, B≈∕(I)0, C-∕(I)γ . . . . (100), where f is a function depending on the nature of the substance. From these equations, by squaring, adding, and extracting the square root, we get ∕(I)∕I = 1∕^, in other words, I, and therefore /(I), are functions of ∣j. Hence we may write the equations (95).

A 1 =F(l)α, B 1 -F(l)3, C 1 -F(l) γ . . . (101). It is easy by means of these to introduce the requisite modifications into the general equations of magnetic equilibrium. For the details we refer our readers to Kirchhoff’s memoir in Grelles Journal, where the matter was first fully worked out. It will be seen at once that the induced magnetization is in general neither solenoidal nor lamellar.

There is one important class of cases in which the conclusions arrived at on the assumption that κ is constant still hold, viz., those in which the induced magnetization is uniform. In such cases I has the same value throughout the body, and κ is therefore constant throughout the body in any one case, although it differs from one case to another. For example, in the case of an ellipsoid the equations (85) above given for the components of magnetization still hold good, provided we understand κ to be defined by the equation

-[K 1 ⅛) ’ ÷(π⅛M) ’ ÷(r⅛)Ι i ]∙<->∙

It is clear, therefore, that by experiments on an ellipsoid placed in a uniform field we could determine the function F(^), and also test the truth of the mathematical theory. For, A 1 ,B 1 ,C 1 being obtained by observation, one of the equations (85) will enable us to determine κ, and the argument can be calculated from α 0 , β 0 , γ 0 and A 1 ,B 1 ,C 1 ; the test of the truth of the theory would be the agreement of the three values of κ obtained from the three equations (85).

Historical Remarks on the History of the Mathematical Theory.— Although the Tentamen of Aepinus, published in 1759, and the discoveries of Mayer and Lambert did much to make clear the exact nature of the problems involved in the modern mathematical theory of magnetism, yet the origin of that theory is usually, and with justice, dated from Coulomb. Not only did the results of his careful and judicious experiments afford the means of bringing a mathematical theory to the test, but the marvellous sagacity he displayed in analysing the phenomena enabled him actually to lay the foundations upon which such a theory could be constructed. After him, Biot and Hansteen, of whose services we have already spoken, are to be reckoned as pioneers. The theory as it now stands was virtually created by Poisson in four of the most admirable memoirs to be found in the whole literature of mathematical physics. In the first two he investigates expressions for the force due to bodies magnetized in any manner; he then applies his formulae to the case of bodies inductively magnetized but having no coercive force. Although he confines his investigations to the case of isotropic bodies, he is quite aware of the general nature of the consequences of aeolotropy, and in fact distinctly predicts as possible the magnecrystallic phenomena afterwards discovered by Pliicker and Faraday. The formulae he gives are practically identical with those given above (p. 248). He works out in detail the solution for the case of a hollow or solid sphere exposed to any system of inducing forces having a potential, and in particular compares the results, when the inducing field is uniform, with the experiments of Barlow. In the second memoir he works out the solution of his equations for an ellipsoid in a uniform field, examining specially the case of an ellipsoid of revolution and its extreme cases (see above, p. 245). At the end of this memoir he discusses the disturbing forces on a compass, arising from the earth’s induction on any distribution of soft iron, and shows that the given components of the disturbing force are expressed by linear functions of the components of the earth’s force, involving nine constants which depend on the quantity and distribution of the iron. The third memoir, on magnetism in motion, is an attempt to explain the phenomena of the deviation of the magnetic needle caused by rotating metal spheres or disks. Although the physical interest of this memoir was in a great measure destroyed by the discoveries of Faraday as to the true nature of this action, yet, as a piece of profound mathematical investigation, this work of Poisson’s is still worthy of study; nor is it perfectly certain that his theory will not after all be required to explain certain residual phenomena. The fourth memoir develops the mathematical theory of the deviation of the compass caused by the iron of ships. After Poisson the most important investigators are Green and Gauss. Green’s services have already been alluded to in the article Electricity; we need only mention here his approximate solution of the problem of the magnetic distribution on cylindrical bars, which gives a formula agreeing with that of Biot. The all-important work of Gauss has already been detailed.

In Crelle ’ s Journal for 1848 J. Neumann worked out the solution of the induction problem for an ellipsoid of· revolution under the action of any conservative system; and six years later, in the same journal, Kirchhoff worked out the case of a circular cylinder of infinite length. We are not aware that the solution of Poisson’s equations in particular cases has been carried any farther, unless we include as new the case of a hollow ellipsoid treated by A. G. Greenhill in the Journal de Physique for 1881.

The most important contributions to the general theory of magnetism since Poisson are to be found in a series of memoirs by Sir William Thomson, the first of which appeared in the Philosophical Transactions for 1851. He divests the theory of Poisson of all particular assumptions connected with the two-fluid theory, and bases it on a small number of principles drawn from observation. He enters more fully than Poisson had done into the specification of magnetic distribution. He gives simple synthetic solutions of the induction problem for spheres and ellipsoids in a uniform field. He gives for the first time with full generality the theory of induction in aeolotropic media, and shows that Poisson’s theory thus fully developed leads to all the laws of paramagnetic and diamagnetic action discovered by Faraday, and also to the laws of magnecrystallic action discovered by Pliicker and Faraday. The value of his theory was fully recognized by Pliicker, and apparently also by Faraday; indeed one of its ablest expositors was Beer the friend and coadjutor of Pliicker. The experimenters who followed these.masters were less intelligent, and the theory of Thomson was for a number of years misunderstood or neglected, the result being much fruitless discussion in which the true issues were often confused. Of late the theory has obtained wide currency and the adhesion of every physicist worthy of the name. Quite recently Thomson’s theory has been further developed in an interesting paper by Helmholtz, chiefly with a view to its application to the phenomena of dielectric polarization.

For the benefit of the mathematical reader we append a list of the more important papers on the mathematical theory of magnetism that have appeared recently, and are not quoted above :—

Plana, “Mémoire sur la théorie du magnétisme,” Ast. Nach., xxxix., 1854; F. Neumann, Vorlesungen uber die Theorie des Magnetismus, delivered 1857, edited by C. Neumann, 1881; Riemann, Schwere, Electricität, und Magnetismus, lectures delivered in 1861, edited by Hattendorf, 1876; Lamont, “Beitrag zu einer mathematischen Theorie des Magnetismus,” Sitzber. d. Bayer. Akad., 1862; I. Weber, Zur Theorie der Magnetischen Induction, Kiel, 1877, see Wied. Beibk, 1878; Rowland, Silli-rnan’s Jour., 1879 (calculation of couple on a body suspended in a heterogeneous magnetic . field); Boltzmann, “Magnetisinιng eines Eisenringes,” Wied. Beibl., 1879; Id., “Heber die auf Diamagnete wirkende Kraft,” Wien. Ber., 1879; Riecke, Wied. Ann., 1881 (approximative solutions of the problem of magnetic induction).

Induction in Strongly Magnetic Bodies. The earliest experiments bearing on the mathematical theory of magnetic induction are those of Barlow and Christie, who determined the deflexion of a compass needle placed in various positions relatively to spheres of cast iron inductively magnetized by the earth’s force. They found that the deflexion α of the compass could be represented by tana = AsinØ cos θ sin φ ∣ r 3 , where J is the angle between the line of dip and the line joining the centres of the sphere and compass, and φ the angle between the plane of these two lines and the plane of the magnetic meridian. It was also found that the deflexion produced by a hollow sphere was as great as that produced by a solid sphere so long as the thickness of the former was not less than the -j-⅜ τ th of its radius.

; All these results of Barlow and Christie are in agreement with the theory of Poisson. Another consequence of great practical importance follows from the mathematical theory, viz., that inside a hollow iron sphere of any considerable thickness the magnetic force is very small in comparison with the external inducing force. Sir William Thomson takes advantage of this principle to render his marine galvanometers independent of external magnetic force by surrounding them with a tube of soft iron.

Along with the experiments of Barlow we may rank those of Pliicker and Dronke as affording us the means of testing the general applicability of the mathematical theory to the magnetization of soft iron. In Plücker’s experiments an ellipsoid of soft iron was fixed in a graduated brass ring with its longest and shortest axes (α and c) in the plane of the ring. When the ring was suspended with the longest axis a vertical in the nearly uniform field between the two flat vertical faces of the poles of an electromagnet, the mean axis b set itself parallel to the horizontal line of force; as the point of suspension was moved along the circumference of the ring a point was reached at which the plane of b and a ceased to set parallel to the lines of force, and the plane of a and c began to do so; ω, the number of degrees between this point and the end of the axis a, was observed. The times of vibration, T α and T c , of the ellipsoid, when suspended so that a and c were vertical, were then observed. By the theory we ought to have tan 2 « = T 2 (δ 2 + c 2 )∕T 2 (α 2 + δ 2 ).

The value of ω calculated by means of T c and T α from this formula was 30° 13'; the value observed was about 29°. The relation connecting T o , T i ,, T c according to the theory is

(α 2 + δ 2 )∕T 2 + (6 2 + c 2 )∕Ti - (c 2 + α 2 )∕T β 2 - 0; and the observed values of T α , T d , T c did, in fact, satisfy this equation very nearly. Dronke’s experiments on ellipsoids of iron and nickel were of a similar character.

Deviation of the Compass.— One of the earliest and certainly the most important of the applications of the mathematical theory of magnetic induction was the discussion of the deviation of the compass caused by the magnetism of the iron in ships. This disturbance seems to have been first noticed by Wales the astronomer, who accompanied Cook on his voyages of discovery (1772 to 1779). The same thing was noticed during the voyage of D’Entrecasteaux in search of La Pérouse; .and Beautemps-Beaupré, who accompanied him, calls attention to the errors thence arising in the surveying of coasts by means of the compass. Flinders, using the numerous observations made by Wales and by himself, endeavoured without success to construct empirical formulae for correcting the errors of the compass. He also attempted to correct the errors partially by means of a vertical bar of soft iron placed near the binnacle. Barlow and Scoresby also occupied themselves with the problem.

The unusually great deviations observed during the Arctic voyage of the “Isabella” and “Alexander” in 1818 attracted the attention of Poisson, and gave rise to his memoir on the subject already alluded to. Important as the matter then appeared, it became still more so after the introduction of iron ships. Investigations both theoretical and experimental were made in England by Johnson, Airy, Evans, Smith, &c. It is to Smith that the mathematical theory as it now stands is mainly due.

The cause of the deviation of the compass is twofold; it arises partly from the permanent magnetism of the ship, partly from the temporary or induced magnetism. The permanent magnetism of the ship is acquired for the most part during the process of building. The earth’s force acts on the iron, and the constant jarring in the process of construction enables it to induce a considerable permanent magnetization, which the ship carries with her to sea. The quantity and distribution of this magnetism will depend greatly on the build of the ship (whether of wood or of iron), and on her position with respect to the magnetic meridian during building. A considerable portion of it is what Airy calls subpermanent; de., it diminishes gradually as the ship is worked. This magnetic settling down will take place more rapidly in a steamer which is constantly agitated by the jarring of machinery than in a sailing ship, unless the latter be subjected to shocks from the impact of waves in rough weather. After a time the ship reaches a more or less stationary condition as to permanent magnetism. Along with the phenomenon of subpermanent magnetism has to be classed what is sometimes called the sluggishness of ships’ magnetism; this arises from the fact that all the temporary magnetism of a ship which has sailed for some time on any one magnetic course in any one latitude does not at once disappear when the course or the latitude is changed, so that to the permanent magnetism of the ship has to be added a subpermanent magnetism depending on her course and position several days before. It is evident that the cause of disturbance at present under discussion is somewhat capricious, and can only be controlled by constant attention on the part of the mariner.

The temporary induced magnetism depends on the ship’s position on the earth, and on her angular position relative to the magnetic meridian; but, so long as the iron in the ship or the position of the compass is not altered, the constants which determine it remain the same; the disturbance can be foreseen, and either allowed for or mechanically corrected with much greater certainty than in the case of the permanent magnetism.

Mathematical Formula for the Deviation.— Let the origin be at the centre of suspension of the compass card; and let the axes of x, y, and z be drawn in the direction from stern to head, in the perpendicular direction from port to starboard, and vertically downwards respectively, the ship for the present being supposed to be on even keel. Let P, Q, R be the components of the magnetic force parallel to these axes arising from the permanent magnetism of the ship; x,y,z the components of the earth’s force; and α∕, y ’ , √ the components of the whole force at the centre of the compass card. Then a!≈x+ax+by+cz + V } ιf≈y+dx+ey+fz+Q ( (103)

z'=z+gx+hy + kz + R) are, according to Poisson’s general theory, the fundamental equations of the subject. P, Q, R are constants depending on the permanent, and α, b, c, d, e,f g, h, k constants depending on the temporary induced magnetism.

By a synthetic process of great interest and importance we may show that the nine constants a, b, c, d, e,f, g, h, k are all independent. For example, if we place a rod of practically infinite length with its end before the binnacle, and stretching forward, or with its end abaft the binnacle and stretching aft, it will give rise to the term ax in x'. If a be negative the rod must be finite and it must run under the binnacle, ending a little fore and aft; again, to represent dx, we must have a pair of infinite rods with their ends to starboard and port of the binnacle, and running fore and aft or aft and fore respectively, according as d is positive or negative; finally, to represent a pair of infinite rods with ends above and below the binnacle, running fore and aft or aft and fore respectively. The reader will have no difficulty in completing the scheme, the rule being that the ends lie in the direction of a/, y', or √, and the lengths in the direction of x, y, or z.

From equations (103) the deviation of the compass is expressed in terms of the magnetic or of the compass course as follows. Let H be the horizontal force of the earth; H ’ the horizontal force of the earth and ship; ∙θ the dip; £ the “magnetic course,” i.e., the azimuth of the ship’s head eastward from magnetic north; ζ' the “compass course,” i.e., the azimuth of the ship’s head eastward from the direction of the disturbed needle; b = ζ-ζ the easterly deviation of the compass. Then

^jsinδ = ^ + ^sinf+Φcosf+ gsin2f+<Ecos2^ )

H <1M) ·

-τjcosδ = l + $cos£- ®sin£+ gcos2f- (gsin2f ) All , , a + e ~ d-b _ ae a , d + b when ∙ λ-l + - r , . S⅛ . «— 5r > s .l(e t, lβ+ ∣), e -l(, ton , + e).

From (104) we get tan δ - ^ + ⅜ s ⅛f+<C0sY+gsin2f+βcos2C

1 + ÿ cos Cφ sin f + g cos2f- ® sin 2f ’ (105) ’ which gives the deviation on any given magnetic course.

From (105) we get by substitution sin δ = ⅜ cos δ + $ sin ζ' + Ceos f + gsin (2f + δ) + « cos(2 f + δ)(l 06), an equation connecting the deviation with the compass course.

When the deviation is not greater than 20° or so, then (106) may be replaced with sufficient accuracy by

3 = A + Bsinf' + Qcos£' + Dsin2£' + Ecos2£' . . (107), where ‰ $, C, C, are nearly the natural sines of A, B, C, D, E.

In the above it is supposed that the ship is on even keel. Strictly we ought to take into account both the pitch and the heel of the ship; in practice the pitch is always so small as to be of no consequence, but the heel, especially in a ship under sail, may be very considerable. When the ship heels through an angle i, the. deviation is obtained from the above formula by writing α⅛ δ 1 ∙, &c., in place of a, b, &c., where ui=a, ¼=δcosi-csini, Ci=ecosi + δsin∕, di=d cosi - g sin i, ei=e-(f+h)cosisini-(e-k) sin a i, fi *-f+ (e - k) cos i sin i - (∕+ Ä) sin a i, gi=gcosi + d sin i, hi=h + {e - ⅛) cos i sin f - (∕ + A) sin 2 ι, ki=k + (∕+ h) cos i sin i + (β - lc) sin 2 i,

Pi = P, Qi=Qcost-Rsini, R,=Rcosi + Qsini.

If the soft iron be symmetrical with respect to the fore and aft central line, and if i he so small that its square may be neglected, then

λ i = λ , 5∣.∙=9, ‰-⅜Λ ^- c -⅛⅛, ΔΛ. ZΛ

(‰=C + -^e- ⅛--^^tan∂f=C +Ji; and if δj represent the deviation for the given compass course ζ ’ when the ship heels i to starboard, δ the deviation on the same course on even keel, then

δi = δ.+ ⅛-^i + Jicosf'-^i¾cos2i ’. . . (108).

∆K ∆K

The part of the deviation which depends mainly on 2( is called the “constant deviation”; it can only arise from horizontal induction on soft iron unsymmetrically placed.

The part depending mainly on $ and βf, viz., Bcos^' + CsinC» is called the “semicircular deviation” because it vanishes and changes sign on two diametrically opposite compass courses, or neutral points. The principal coefficient of the semicircular deviation is ⅛J = (ctan0 +P∕H)∕λ; ctanØ/λ arises from vertical induction in soft iron before or abaft the compass; P∕λH arises from the permanent magnetism of the ship. The second coefficient φ = (∕tan0 + Q∕H)∕λ consists of ∕tan0∕λ, arising from soft iron unsymmetrically placed, and therefore in general very small, and Q∕λH arising from permanent magnetism. ⅛) can be reduced to zero by a magnet placed fore and aft with its centre in a transverse vertical plane passing through the compass, ® by means of a transverse magnet in a fore and aft plane through the compass.

In wooden ships the courses for which the semicircular deviation vanishes are nearly north and south; but in iron ships they approximate to those points of the compass towards which the stem and stern lay in building.

The terms Dsin 2ζ ’ + Ecos2^', depending mainly on the constants ,51 and C, are called the “quadrantal deviation.” This part is alternately easterly and westerly in the four quadrants, vanishing on four compass courses. § = (α— e)∕2λ is the principal coefficient of the quadrantal deviation; it depends on horizontal induction in symmetrically placed fore and aft or transverse soft iron. It is in general positive, and in that case can be reduced to zero by two transverse rods with their ends symmetrically placed to starboard and port of the compass. In practice two hollow spheres an inch or so thick are used instead of the rods. The other coefficient ® = (d + δ)∕2λ is in general small, as it depends on horizontal induction in soft iron unsymmetrically placed. It is only when the ship heels that this coefficient is in general of any importance.

.Whereas the semicircular deviation depends both on the geographical position of the ship and on the state of its subpermanent magnetism, the quadrantal deviation is independent of both, and can be corrected mechanically once for all, or allowed for by means of tables constructed from observations made in any one place. The amount of the semicircular deviation in England does not exceed 10° for wooden ships of war, but in iron-built ships it frequently exceeds 30° even at the standard compass. o The quadrantal deviation in wooden ships does not often exceed l o or 2°; in ordinary iron ships it ranges from 3° to 7°, but in some armour-plated iron, ships of war it has reached as much as 8⅜ o at the standard compass, and 15° for compasses less favourably placed.

The chief part of the heeling deviation is the term Jzcosf', depending on the coefficient J = (e- ä - R∕Z)tan0∕λ. This coefficient may be reduced to zero by increasing or diminishing the earth’s vertical force by means of a vertical magnet under the compass.

The usual way of ascertaining the deviations of a ship’s compass is to “swing” the ship gently round so that her head comes into various positions, and to observe with the compass the magnetic bearing of some well-defined distant point (compass mark) on shore. The true magnetic bearing of this point is then ascertained, which may be done by taking the compass ashore, carefully placing it in a line joining the compass mark with the point on board at which the compass was formerly placed, and then taking the magnetic bearing of the mark once more. Care must of course be taken that there is no local magnetic disturbance at the shore station. The differences between the bearings on board and the bearing on shore give of course the deviations for the various positions, of the ship’s head.

When the deviations have thus been ascertained they may be either corrected by means of tables, by graphical methods, such as the steering diagram of Napier or the dygograms of Smith, or mechanically as we have partially explained. For full details on the subject the reader should consult the Admiralty Manual on the Deviation of the Compass.

Of late years Sir W. Thomson has devoted his great scientific knowledge and well-known practical sagacity and inventive skill to the improvement of the compass. By reducing the size of the magnets and increasing their number he has succeeded in reducing Airy’s apparatus for the mechanical correction of the quadrantal deviation within convenient bulk, and by lightening the card and suspension of the magnets in a very ingenious manner (at the same time throwing all the remaining weight as much as possible to the circumference) he has reduced the friction on the pivot to a minimum while retaining a sufficiently long period of vibration to secure perfect steadiness. He has also contrived apparatus for facilitating the determinations of the deviation on different courses and of the heeling error.

The experimental investigation of induced magnetism reduces itself mainly to the investigation of the dependence of the magnetic susceptibility κ (or the magnetic permeability ττ) upon the magnetizing force ∣j. Confining ourselves to the strongly magnetic metals, iron, nickel, and cobalt, it will be seen presently that κ depends, not only upon ∣j, but also upon the magnetic condition of the body at the actual moment when is in action, and upon its previous magnetic history, κ also depends greatly on the temperature, on the state of the body as to purity (notably in the case of iron and steel on the percentage of carbon present), and on the temper. Thus, if we make one experiment on a body by magnetizing it in any way, we permanently alter its magnetic properties, and can restore it to the magnetically virgin condition only by heating it to a high temperature; but in this process we are very apt to permanently alter its molecular condition, so that, although magnetically indifferent, it is physically changed. Owing to the fact, already insisted upon, that we cannot infer the magnetic distribution inside a heterogeneously magnetized body from its external magnetic action, and to the fact, presently to be established, that κ varies with %j, it is of the last importance to choose the experimental circumstances so that both the magnetic field and the induced magnetization shall be uniform, or very approximately so. A further necessity for the fulfilment of these conditions arises from the fact that we must in all cases be able to render an account of the effect of the form of the magnetized body, because the true argument of κ is not the strength of the original field but the whole force ∣j due to the original field and the induced magnetism together.

The simplest method for securing a uniform field whose strength can be controlled is to place the body inside a hollow cylindrical coil (usually called the magnetizing spiral), whose length so far exceeds that of the body that the disturbance arising from the ends of the coil may be neglected in the neighbourhood of the body. The results in all cases where the length of the body or core is nearly equal to or exceeds that of the coil are impure, and can only be used with the greatest caution in drawing general conclusions as to the value of κ. The core should always be either exactly or approximately one of the calculable forms, but preferably such that the dimension parallel to the axis of the spiral very much exceeds the others, because in this case the effect of the form is of secondary importance compared with the effect of the susceptibility (see above, p. 245). Thus a very thin cylindrical core is convenient, because the force inside it differs very little from that of the undisturbed field, and any small difference can be easily calculated by supposing the cylinder replaced by a very elongated ellipsoid. On the other hand, a thick-cylindrical bar is a bad form of core for the determination of κ, both because the magnetizing force inside it is less than the intensity of the undisturbed field by a large quantity, which it is impossible to calculate, and because the magnetization at the end is not uniform, and the disturbance thereby arising is so great that it may mask the general character of the function κ altogether. A further question arises as to how far the time during which a magnetizing force acts affects the resulting magnetization, whether temporary or permanent. It is also important to consider the disturbances arising during the make and break of the current in the magnetizing spiral. As the resistance in the circuit is usually small, and the self-induction and capacity sensible, oscillatory currents may arise; to these will correspond oscillatory magnetizing forces, which may even vary in sign. When we consider that the permanent magnetization produced by any force may be very much weakened or even altogether destroyed by a smaller force in the opposite direction, it is evident that we have no right to conclude that these disturbances, especially at break, will be without effect upon the permanent magnetization. In order to elude these difficulties, some experimenters have followed the practice of first establishing the current, then gently introducing the core into its place, and finally removing it before breaking the circuit. In this way the disturbances just alluded to are avoided; but another difficulty is raised, for it is clear that in this operation the core passes through a heterogeneous field before it reaches the final position where the magnetizing force is uniform; different parts of it have therefore been subjected successively to different influences, and we are not at liberty α priori to conclude that this fact will not influence the results. Perhaps the best plan would be to place the core iu its position, and allow the current to rise very slowly to the maximum value required, and then to fall slowly to zero. This, however, is not the place to dogmatize concerning the best method of experimenting; all that is necessary is to furnish the reader with points of view from which to criticize the experimental results now to be cited.

In the researches of Lenz and Jacobi the magnetic moment of the core was measured by the induction current in a secondary coil placed upon the magnetizing spiral. A considerable portion of their work was directed to proving principles which we here take for granted, e.<7., that the magnetizing force is independent of the thickness of the wire of the magnetizing spiral, of the radius of its windings, and so on. They concluded from these experiments that the magnetization is proportional to the magnetizing force ; i.e., κ is constant for a given quality, &c., of metal. The experiments of Joule, which were made independently about the same time, led in general to a similar result. His method consisted in measuring by means of a balance the attraction P between two electromagnets actuated by the same current C. If the magnetization of the core were strictly proportional to the magnetizing force, i.e., to the current, then P would be proportional to C 2 , and P∕C 2 would be constant. In most cases this was so; but in two cases, where the cores of the electromagnets were very thin and the windings more than usually numerous, the ratio P∕C 2 was found to decrease as the current increased. This shows that the magnetization tends to a maximum value as the current increases, in other words, that, for very large values of ∣), κ decreases.

Müller, using the method of deflexions, arrived at a similar conclusion. His cores were 56 cm. long and from 9 mm. to 44 mm. thick, his magnetizing spirals from 48 ’ 2 cm. to 53 ’ 2 cm. long; his results are therefore impure, and the empirical formula by means of which he represents them of comparatively little importance; but the approach to a maximum of magnetization (saturation) is quite clearly demonstrated. He found, in accordance with theory, that if we increase the external magnetizing force (^ 0 ) saturation is more quickly reached in thin than in thick bars. Somewhat similar experiments were made by Von Waltenhofen, who deduces from some of his own experiments with very thin cores, and from the experiments of Müller, Weber, and Dub, 1678 to 2125 mm. mg. sec. units of magnetic moment per mg. of iron as the maximum of magnetization. This would give from 1317 to 1668 C.G.S. units for the maximum magnetic intensity in iron. These numbers, derived from more or less impure results, are merely rough approximations, but they agree very well with those derived at a later date by methods less open to theoretical objections.

The approach to saturation may be very neatly demonstrated as follows. The same current is sent through a galvanometer and through the coil of an electromagnet with a thin core. The electromagnet is so placed that its action on the needle of the galvanometer just compensates the action of the galvanometer coil for a particular strength of current; the needle then points to zero. If now the current be increased, since the increase of magnetization does not keep up with the increase of the current, the action of the coil prevails, and the needle deviates accordingly.

The most extensive and important of the earlier researches into the general nature of magnetic induction are those of Wiedemann. An epitome of his results, with references to contemporary or preceding researches in the same direction, will put the reader in possession of almost all the more important general facts known until the quantitative experiments of Stoletow, Rowland, and their followers gave a complete account of the general characteristics of the function κ.

In these experiments the method of deflexion was used. The magnetizing spiral was placed magnetic east and west, and in the continuation of its axis was hung a magnetic steel mirror in a thick copper box to damp its oscillations. The deflexions of this mirror, read as usual with a scale and telescope when the core was not in, gave a measure of the current; and the increase of the deflexion on introducing the core gave a measure of the magnetic moment of the core. The cores were cylinders 22 cm. long., 1 ’ 35 cm. thick, and the length of the spiral was only 24 cm.,—so that perfectly pure results could not be obtained. To compensate to some extent for the shortness of the spiral, the bars were gently drawn to and fro several times before being placed in the final position· for which the reading was taken. In order to measure the permanent magnetism the core was removed, the current broken, the core returned to its former position, and a reading again taken. The conclusions arrived at were as follows.

I. When a steel or iron bar is magnetized for'the first time by a current C, the temporary moment K produced during the action of the current at first increases faster than the current, then more slowly, and finally tends to a maximum, as shown by Joule and Müller. The period of quicker increase is more marked in long than in short bars; it shows itself even on remagnetizing bars that have been several times magnetized and demagnetized. As C increases, the maximum of K is reached sooner in thin and long bars than in short and thick bars. Between the period of increase of K/C and its period of decrease there is no period of any considerable length for which it is constant. This last fact may be shown by means of the experiment of Koosen described above; viz., if the compensation be made for very small currents, when the current is increased, at first the electromagnet prevails, and the needle goes to one side of zero, then the current in the coil prevails, and the needle returns towards zero, and finally deviates on the other side.

The point at which the ratio K/C has its maximum for any particular electromagnet is called by Wiedemann the “turning point” (Wendepunkt). The turning point relates to the body as a whole, and the value of the external magnetizing force ∣p 0 for which it occurs depends both on the form of the body and on the nature of the metal. It has therefore no very definite physical meaning. It must be carefully distinguished from the “saturation point.” Any element of a body is said to be magnetized to saturation when no increase of the magnetic force can increase its magnetization any farther. It may happen, however, that some parts of a body are magnetized to saturation while others are not. With regard to the turning point, Dub has shown that with similar and similarly wound cores the turning point occurs for the same value of the current. This is of course in agreement with an obvious corollary of the general theory of magnetic induction.

II. In a freshly magnetized bar the permanent moment which remains after the action of the current has ceased at first increases quicker than the producing current; but for stronger currents a turning point is reached; and then the moment increases more slowly than the current, and approaches a maximum.

III. In attempting to destroy the permanent magnetism of a bar by means of a demagnetizing current, it may happen that a current, which, during its action, already produces a temporary magnetic moment of opposite sign, still leaves on ceasing to act a permanent magnetic moment of the same sign as before, although less in amount. On increasing the demagnetizing current still farther the permanent moment is at last destroyed. In this process the permanent magnetism decreases faster than the demagnetizing current increases,—so that the current required to destroy a given permanent magnetism is less than the current that originally produced it.

IV. When a fresh bar has been magnetized with any permanent moment, and then demagnetized by a current

- C', opposite to the magnetizing current C, a second application of - C', or of any weaker current in the same direction, will not produce a reverse permanent moment, although a current C' in the same direction as C will magnetize the bar permanently in the original direction more or less strongly. It follows therefore that demagnetizing by an opposite magnetic force, although it may destroy the permanent magnetism of a body, does not render it magnetically indifferent, as heating to a white heat would do. The body remains in fact more easily magnetizable in one direction than in another.

V. In certain cases a fresh bar was magnetized by a current C, and then partly demagnetized; it was then found that a current C was required to bring it back to its original permanent moment.

VI. In another case a fresh bar was magnetized by a current C to permanent moment K, then reduced by a demagnetizing current C' to permanent moment K', then by a direct current C" less than C brought to permanent moment K". It was then found that a current C' was necessary to bring it back to permanent moment K'; and this held whether K' was positive, zero, or negative.

VII. When a bar is repeatedly magnetized and demagnetized by currents of the same intensity, the permanent magnetic moments corresponding to a given force become, to begin with, a little greater than at first; to begin with, they increase faster than the magnetizing force, though not so fast as at the first. The turning point, however, occurs for a weaker current than before. The magnetization obtained with the strongest current gradually decreases a little. The moments left by the demagnetizing current decrease less rapidly than before, so that a current at first capable of demagnetizing the bar altogether leaves after repeated magnetization and demagnetization a slowly increasing residual moment. After a large number of repetitions of the operation of magnetization by a current C and demagnetization by a current

— C', the bar finally reaches a constant state, so that each magnetization and demagnetization leaves a corresponding invariable permanent moment. When we pass beyond the limits C and - C', these phenomena are repeated in the same order as before.

VIII. All the above phenomena are most clearly seen in hard steel, less clearly in soft steel and iron. For small magnetizing forces the temporary moment in hard steel is less than in soft steel, and greatest of all in soft iron. The general rule is, the harder the material the less the temporary and the greater the permanent moment for a given magnetizing force.

IX. If, however, we consider the ratios of the temporary moments in soft steel and iron to the temporary moment in hard steel, all for the same force, then these ratios decrease gradually as the force increases: so that the tem porary moment in soft iron reaches its maximum sooner than in soft steel, and still sooner than in hard steel.

The earliest experiments from which definite values of κ have been calculated are those of Weber. A cylindrical bar, 10Ό2 cm. long and - 36 cm. thick, was placed inside a spiral so long that the magnetizing force throughout the length of the bar could be assumed to be uniform. The moment of the bar was measured by the method of deflexion, the action of the spiral on the deflected magnet being compensated by means of a part of its own circuit suitably arranged. The intensity of the current, in the spiral was found in absolute measure by means of a tangent galvanometer. Assuming that the bar could be replaced by a very elongated ellipsoid, Kirchhoff calculated by means of the theory explained above (p. 249) the values of κ for values of g) ranging from 29 ’ 6 to 248 ’ 4 (C.G.S. units), and found that it decreased steadily from 25Ό to 5 ’ 6. In the experiments of Von Quintus Icilius bars were used which had been reduced by filing as nearly as possible to the form of ellipsoids of revolution. The magnetic moments were measured partly by the deflexion method, partly by the method of electromagnetic induction. In this last method a secondary spiral is placed upon the magnetizing spiral, and the induced current in it, caused by reversing the magnetizing current, is observed first when the ellipsoid is in the magnetizing spiral, secondly when it is not. When these currents are known in absolute measure, the moment of the ellipsoid can be calculated. The experimenter did not himself reduce his results so as to obtain κ, but contented himself with remarking that the ratio of the whole moment of the ellipsoid K to the strength of the undisturbed field g) 0 reached a maximum as g) 0 was increased, this maximum occurring for smaller values of gj 0 the more elongated the ellipsoid. The true meaning of his results was brought out by Stoletow, who reduced them, and established the interesting fact that, as the magnetizing force g∣ increases from very small values, κ at first increases rapidly, then reaches a maximum, and afterwards decreases more slowly. For one ellipsoid κ increased from 30 ’ 5, for g∣ = ∙24, to a maximum 120 ’ 4, for ^j = 4 ’ 56, and then decreased to the value 39 ’ 4, for g∣ = 30O7. In another, the initial value was 20 ’ 1 for g∣ = ∙518, the maximum value 107 5 for g) = 4 , 92, and the final value 2 ’ 86 for g) = 454 ’ I.

Thalén, adopting a method indicated by Weber, determined the value of κ for small magnetizing forces. Long bars were placed in the axis of a cylindrical coil considerably exceeding them in length. This coil was caused to rotate 180° about a horizontal-axis, so that the magnetization induced by the earth’s vertical force was reversed relatively to the coil. The current thus caused was meaured by means of the swing of a galvanometer in circuit with the coil; from this (see above, p. 240) the moment of the induced magnetism was calculated; and thence, assuming the bar to be replaceable by an ellipsoid, κ was calculated. From three bars of the same metal each 400 ’ 4 mm. long, having diameters of 36 ’ 4, 29 ’ 94, and 23 ’ 87 mm. respectively, the values of κ deduced were 32 ’ 32, 3D80, and 32 - 64. For other specimens of iron he found values of κ ranging from 27 ’ 24 to 44 ’ 23.

A set of observations on ellipsoids of revolution were made by Riecke, by the method just described. The ellipsoids, seven in number, were all cut from tl⅛e same piece of soft iron, but varied in volume and in eccentricity. The resulting values of κ were found to be independent of the volume of the ellipsoids and of the part of the iron from which they were cut; but, on the other hand, with one slight exception, they increased with the eccentricity of the ellipsoids. Kohlrausch, in communicating these results to Poggendorff ’ s Annalen, remarked that they stand in contradiction with the theory of Poisson and Neumann; in so saying he probably considered the constant vertical force of the earth (∣j 0 ) to be the argument of the function κ; but this is not so, as Stoletow points out in the paper already quotêd. The actual magnetizing forces are greater in the more elongated ellipsoids; and Riecke’s results simply prove that for values of ∣j varying from Ό31 to ’ 072 κ increases from 13 - 5 to 25 ’ 4.

In order to establish the initial increase of the magnetization function κ beyond all doubt, Stoletow (Z. c.) made a new set of experiments on a carefully annealed iron ring of rectangular section (exterior diameter 20 cm., interiordiameter 18 cm.,height D47 cm.). The ring was carefully wound throughout with a primary coil of η ( = 800) windings; over this, in one or more shorter or longer stretches, was wound a secondary coil of n'( = 50 to 750) windings. The induction current in the secondary, due to the reversal of a known current i in the primary, was sent through a galvanometer, and thus measured. If E be the electromotive force of this current, then (see above, p. 246) E = 4n√i(4τrκM + P), where M and P can be calculated from the dimensions of the ring and its primary coil. All then that is necessary is to know E/» in absolute measure. We refer the reader to the original paper for the details of the measurements. The results are very interesting, and fully confirm the conclusions drawn from the results of Von Quintus Icilius and Riecke. The smallest value of ∣j was ’ 43, and the corresponding value of κ 2D5; the maximum value of κ was 174, for ∣∣ = 3∙2j the last value observed was κ = 42 ’ l, for ⅛ = 30 ’ 7. The temperature varied from 15° C. to 20° C., but it appeared from the experiments that κ did not alter much for moderate changes of temperature. In figure (34) is given a transcription of the curve that represents the results of

Stoletow’s experiments; the abscissae represent the values of ∣} in C.G.S. units, and the ordinates the corresponding values of κ.

About the same time as Stoletow, and independently, Rowland made a much more extensive series of experiments, the results of which form one of the most important contributions yet made to our knowledge of magnetic induction. The experiments were made partly on very long bars; but the published results were mostly obtained from rings, it having been found that the effect of the ends of the bar was sensible even when the length was as much as 144 times the diameter. About a dozen rings of iron, nickel, and cobalt were used; the section was circular in all cases; and a primary and a secondary coil were used as in Stoletow’s experiments. The primary current was measured by means of a tangent galvanometer in which 1, 3, 9, 27, or 48 coils could be brought into operation according to the sensibility required. The induction current in the secondary was measured by the swing of a Thomson’s galvanometer fitted with a heavier needle than usual. The indications of this last were reduced to absolute measure by taking the swing caused by turning over a horizontal coil of known area, inserted in its circuit, so as to produce the full induction current due to the earth’s vertical force. In order to obtain the total induced magnetization the primary current was reversed. To obtain the permanent magnetism it was simply broken; this gives the part of the induced magnetism that disappears with the inducing force (temporary magnetism Rowland calls it); subtracting this from the total magnetization, we get the permanent magnetization. Care was taken in these experiments always to work with magnetizing forces of ascending magnitude, as it was found that the effect of any force is considerably modified if a greater force has previously acted on the body,—in other words, that the magnetic permeability of iron or steel is much affected by pre-existing permanent magnetism. This fact raises an objection to the ring method; for permanent magnetization in a ring is not easily discoverable, and would give it a one-sidedness, so that a magnetizing force would produce much more alteration when exerted in one direction than it would when exerted in the other. Rowland publishes about thirteen different tables, relating to rings of iron and steel in different states, and also to nickel and cobalt, under different conditions as to temper.

magnetization, and so on; the results for cobalt are, however, held to be less satisfactory than those for iron and nickel, for a variety of reasons which he assigns.

In treating his results graphically, two methods are followed. In the first the magnetic induction ∣J is plotted against the magnetizing force ∣) as abscissa. Figure 34 shows the curve obtained in this way from one of his tables. In the second method (1) the permeability ∏r is plotted against the magnetic induction or (2) the susceptibility κ is plotted against the intensity of magnetization $. Either variety of the second method leads to a curve having the general form shown in figures 35 and 36.

The curves obtained, whether for ≡∙ and or for κ and $, fall very rapidly* and ultimately to all appearance almost straight, towards the axis of or $. This suggests that or $ or both reach a maximum when ∣) is increased indefinitely. Supposing such an increase of ∣) possible, the question arises as to which it is that actually reaches a maximum. Most experimenters seem to assume that $ does so, but it must be remarked that this is simply an assumption. Several delicate points of great physical interest might be discussed here, but it will be sufficient to refer the reader to the introduction to Rowland’s second paper.

The general conclusions to be drawn from these experiments are as follows :—

1. The magnetic properties of iron, nickel, and cobalt at ordinary temperatures differ in degree but not in quality.

2. As the magnetizing force ∣j increases from 0 upwards, the permeability of iron, nickel, and cobalt increases until it reaches a maximum, and. after that diminishes down to a very small value. The maximum value is reached when the metal has attained a magnetization of from - 24 to ’ 38 of the maximum. The following table will give some idea of the order of the magnitudes involved; ≡r 0 denotes the permeability for = 0, √ the maximum permeability, and ∣)' the force for which it occurs. In some cases the actual maximum is given, in other cases simply the greatest recorded in the tables of experimental results, and the values of ^) are stated roughly; strict accuracy is of no consequence, owing to the great variability of all the magnitudes. [table]

The smallest permeabilities (for large forces) observed were—for iron 258, for = 64; for steel 246, for ¾j — 48; for nickel 41, for = 131; for cobalt 55, for = 147.

3. The curve showing the relation between κ and $, oι between τr and is of such a form that a diameter can be drawn bisecting chords parallel to the axis of· $ or 3β, and its equation is approximately _ . f x + by+d∖ 3∕ = Bsιn 5 — J , where y = κ or «·, # = $ or ¾l, and δ,<Z, B,D are constants.

4. If a metal is permanently magnetized, its permeability is less for low magnetizing forces, but is unaltered for high magnetizing forces. This applies to the permanent state finally attained after several reversals of the magnetizing force; but if we strongly magnetize a bar in one direction, and apply a weak magnetizing force in the opposite direction, the change of magnetization will be very great.

5. Iron, nickel, and cobalt all probably have a maximum of magnetization, although its existence can never be entirely established by experiment, and must always be a matter of inference. If such a maximum exists, then at ordinary temperatures it will be roughly as follows :— [table]

6. The permeability of any metal depends on the quality of the metal, on the amount of permanent magnetization, on the total magnetization, and on the temperature.

7. The permeability of nickel and cobalt varies very much with temperature. In nickel for a moderate amount of magnetization the permeability increases with rise of temperature, but for high magnetization it decreases. This is very well shown in fig. 36, where the permeability curves for 15° C. and 220° C. intersect each other. In cobalt, on the other hand, the permeability appears to be always increased. The permeability of iron is not much affected by moderate changes of temperature.

8. The maximum of magnetization of iron and nickel decreases with rise of temperature, at least between 10° C. and 220° C., the first very slowly, the second very rapidly. At 220° C. the maximum for iron is $ = 17200 or ∣ = 1360, and for nickel $ = 4900 or ∣ = 380.

The researches of Stoletow and Rowland have undoubtedly made clear the main phenomena of magnetic induction; but in so doing they have raised a host of other questions which have not as yet been settled. There is no lack of recent work bearing on them, but it would be a difficult matter to give succinctly a complete account of the conclusions arrived at. The results of the different experimenters are not seldom contradictory, and the circumstances of experiment are often so complicated that criticism with the view of reconciling them seems hopeless in the meantime. While, therefore, we shall give a fairly complete list of the literature, the reader must not expect in this article an exhaustive analysis of the different memoirs that have recently appeared. Any remarks we shall make have chiefly for their object to call attention to the prominent questions that have been raised by the different workers.

Riecke made a series of experiments on ellipsoids of soft iron; he expresses his results in terms of p the magnetization function for a sphere, and finds, as he ought to do, that, for a considerable range of values of the magnetizing force, p is approximately constant. In point of fact this method of representation is bad, for the quality of the metal only begins to affect p about the fourth or fifth decimal place. Similar experiments on spheres and ellipsoids of soft iron were made by Fromme; and a very extensive series by A. I.. Holz on ellipsoids of iron and steel, in which he gives tables and curves showing the values both of p (to a large number of decimals) and of κ; and the values of the temporary, permanent, and vanishing magnetisms for a considerable range of magnetizing forces. The results, although wanting in regularity and smoothness for the harder kinds of steel, agree in the main with those of Stoletow and Rowland. Holz enters largely in this and in a former paper into speculations concerning the effect of the molecular structure of the metal upon its magnetic properties.

Relating more particularly to the phenomena of the permanent and temporary magnetization of steel we have important memoirs of recent date by Bouty, Fromme, and Auerbach. Bouty’s papers, besides copious references to the general literature of the subject and interesting critical discussions of magnetic theory, contain the results of careful investigations as to the permanent magnetization attained by repeated applications of magnetic force under various circumstances, and verifications of the formulae of

Green for the magnetic distribution in thin needles and cylindrical bars of steel. Two points as to his methods are worthy of notice. He employs a very simple method of measuring the magnetic moment of small pieces of steel: a small needle of moment m attached to a stiff stem, which carries a mirror, is freely suspended and allowed to come to rest in the magnetic meridian; the needle whose moment x is to be measured is then inserted into a tube fixed to the stem with its axis at right angles to the former needle. The deviation α of the compound system being measured by means of the mirror, we have x = m tana. He studies the magnetic distribution in very thin hard needles by the method of rupture, finding that, if the needle be carefully broken, so that the distortion or shock caused by the bending does not extend far from the point of rupture, the magnetic moment of the different parts is little, if at all, affected. For thicker magnets he uses the ordinary method of deflexion.

Bouty found, in agreement with Hermann Scholz and Frankenheim, that, although the continued application of a magnetizing force does not increase the resulting permanent magnetization, the repetition of its application wilL He finds for the magnetic moment y of a thin needle passed x times through a magnetizing spiral the formula V = A - B∕x, where A and B are constants: e.g., in one case, A = 57*78, and B = 6 ’ 32. The ratio A∕(A-B), that is, the ratio of the moment attained by an infinite number of applications of the magnetizing force to that attained by one, decreases as the force increases; on the other hand, if R' be the force required to produce by a single application the same effect as R produces by an infinite number, he finds the ratio R'/R fairly constant (viz., from 1Ό60 to 1Ό65 in his best experiments) for values of R ranging from 10 to 42. In certain cases where the magnetization was effected by induced currents, he finds the formula y = A + B (l-e~ ax ) to represent the results better.

He found that Green’s formula,

. „/ 2 e iβ χ -e~iβ χ ∖ v -λa-[x - - s where B--∙

α

giving the moment of a cylinder of length x and diameter a, was sufficiently accurate both for temporary and for permanent magnetism, and for hard or soft tempered steel, whether saturated or not, provided the bars were in a virgin condition before magnetization. For example, in a saturated bar of soft steel (α = 7 mm.), for the temporary magnetism A = 4O81, B = 1/7442; for the permanent magnetism A = 2*34, B = 1/17'857. In a non-saturated bar of soft steel (α = 10 mm.), for temporary magnetism A = •9966, B = 1/7’142; for permanent magnetism A= , 723, B = 1/17*857; so that B is independent of the magnetic force. With hard tempered bars, A was less, both for temporary and permanent magnetism, than with soft bars; B was independent of the magnetizing force for temporary magnetism, but increased for permanent magnetism with large magnetizing forces. He calls the magnetic distribution long or short according as B is small or great, and explains the phenomena of demagnetized or remagnetized bars by the superposition of long and short distributions. His final conclusionis that there is a greater independence between permanent and temporary magnetism than is usually admitted; and he starts a theory that magnetic bodies are composed of a mixture of two kinds of magnetic molecules, one kind retaining all the induced magnetism, the other wholly devoid of coercive force.

It is obvious, from the results of Wiedemann, Frankenheim, and Bouty just alluded to, that the assumption made in the mathematical theory, that the effect of a magnetizing force is independent of the previous magnetic history of the body, is not even a first approximation to the actual truth. It becomes a matter of importance therefore to study the modification in the induced magnetism corresponding to any force produced by the forces that have preceded it. This effect has been called by German experimenters the magnetic after-effect (Magnetische Nachwirkung). Fromme and Auerbach have recently occupied themselves with this subject, and it may be of some interest to the reader to indicate a few of their conclusions.

In his first paper Fromme experiments with rotational ellipsoids of soft steel, using partly the method of Weber, Thalén, and Riecke, partly the ordinary method of deflexion. He found, in the first place, that the generalized theory of magnetic induction was applicable for values of ∣j varying from Ό061 to ’ 132, κ decreasing between these limits from 23’5 to 8 ’ 68. He attempted to find the maximum force for which permanent magnetism first appears, and fixes it with some reserve at from *2 to ’ 3. The curve which he indicates for the temporary magnetization of soft steel has two points of inflexion, being first concave to the axis of ∣j, then convex, and finally concave again.

He confirms the observation of Frankenheim that repeated applications of the magnetizing force increase the permanent magnetization up to a certain limit, and finds that when that limit is reached the body behaves towards all smaller forces having the same direction as if it were devoid of coercive force. Experimenting on ellipsoids permanently magnetized in this way, he found the mathematical theory of Kirchhoff to be inapplicable, it being impossible to fit the results obtained with the different ellipsoids together; and the discrepancy was greater with the softer than with the harder steel. For forces that are not sufficient to alter the permanent magnetization, κ decreases with decreasing force, as is the case with soft iron, so long at all events as the forces are not very great; and, again, for such forces the variation of κ is more regular the greater the permanent magnetization.

The number of impulses required to saturate with permanent magnetism was greater the greater the ratio of the moment of saturation to the initial moment, e.g., greater for hard than for soft steel. It was found, in extension of a result of Frankenheim’s, that, if U be the original moment, R 1 that produced by one and R that produced by an infinite number of impulses of the magnetizing force, then (U + Rj)∕(U + R) is tolerably constant; but R 1 ∕R decreases with increasing magnetizing force.

With reference to the non-permanent magnetism of a bar repeatedly magnetized by the same constant current, he concludes from his researches that it diminishes, but in such a way that the total induced magnetism remains constant,—so that what is lost in non-permanent is gained in permanent magnetism.

In his second paper Fromme experimented both with iron and with steel cylinders, pointed at the end, of lengths varying from 140 to 220 mm., and of thicknesses from 1 ’ 5 to 8 mm. The method of deflexion was used, the effect of the magnetizing spiral itself being compensated by an auxiliary spiral suitably placed. The cores were carefully introduced into the spiral after the current was established, removed before it was broken, and then replaced when the permanent magnetism was determined.

In the following extract from his conclusions T n denotes the total induced magnetization, R n the whole residual or permanent magnetization, V„ the non-permanent or vanishing magnetization, after n impulses of a given magnetizing force, the suffix being dropped when the number of impulses is not in question, and replaced by ∞ when the number is so great that by further increasing it no alteration in the effect is produced.

A constant force greater than all preceding induces a T which varies with successive impulses, sometimes increasing, sometimes decreasing. If a bar previously heated white hot be subjected to a large force, successive impulses usually give a decrease of T. If, however, the force is preceded by one somewhat smaller, successive impulses usually give an increase. It depends merely on the magnitude and the number of impulses of the preceding force P whether the repeated impulses of a force p will give an increasing or a decreasing T.

R always increases with successive impulses until the limit is reached, and always faster than T; hence increase of R and decrease of V go hand in hand; the magnitude of this increase depends on P and p, and approaches zero with P- j o.

In order that the action of a force p may not be influenced by the after-effect of smaller forces preceding it, it must be applied so often that its further application ceases to increase R. When saturation for R is thus reached, then T, R, and V have the values corresponding to frequent impulses of p for a fresh bar.

R 1 ∕R 00 , R 2 ∕R o o » &c., Ml starting from unity, decrease as the force p increases from zero, diverging more and more until they all reach minima for the same value of p; they then converge again towards unity, which they all reach at the maximum of permanent magnetization. The values of p corresponding to the maxima of R oo ∕p, . . . . ^Rfp, R 1 ∕p are in ascending order of magnitude, and the first of them is the value corresponding to the minima of R 1 ∕R 00 , R 2 ∕R oo , &c.

What was stated for R 1 ∕R 00 , R 2 ∕R 00 , &c., holds word for word for T oo ∕T 1 , T oo ∕T 2 , &c. Hence the decrease of T is conditioned solely by the increase Of R; so that it would appear that the after-effect of a preceding force P depends on the R which it produces. It would therefore be more correct to say that the after-effect depends on r -- R than to say that it depends on p - P.

When a bar has been magnetized by any force P, all smaller succeeding forces leave R unaltered, yet by repeated impulses of p (<P) T decreases until it reaches a certain limit. We may repeat the process as often as we please by always beginning with a new application of a larger force P; if we vary P, keeping p constant, T 1 , T 2 , &c., vary, but the limit T 00 is always the same. In these experiments it was indifferent whether a few seconds or several hours elapsed between the applications of P and p; time had no influence on the vanishing of this species of magnetic after-effect. On the other hand, several impulses of the greater force gave no more after-effect than a single impulse, of whatever duration. If N denote the after-effect of a greater force P upon the action of a smaller p, the law of the phenomenon is

N =cp β (P-j>) i , where c is a constant and a and δ are constant positive numbers, b being a proper fraction, and a possibly very near unity. This of course gives N = 0 for p = 0 and for ρ = P, and gives a maximum value of N for some value of ρ between 0 and P.

The interposition of a force P' between P and p increases the after-effect if P'>P, diminishes it if P'<P; and this holds irrespective of the sign of P' - p.

If we denote by k the susceptibility of a body for vanishing magnetism (V) induced by any force ρ, the question arises how far this is influenced by the permanent magnetism R induced by preceding greater forces. Jamin holds that k is approximately, and Chwolson that it is absolutely, independent of such permanent magnetism. Fromme finds that, when a force ρ, capable of itself producing a permanent magnetism r, acts on a bar already possessing a permanent magnetism R>r, then k is increased (by the presence of R) if R - r is small, but diminished when R - r is great. The after-effect for small forces ρ may therefore be either increase or decrease of k ; but for large forces ρ it is always increase.

At the conclusion of his paper Fromme points out the contrast between magnetic and elastic after-effect, and dwells upon the analogy between his results and those of Thalén concerning the limits of elasticity in solid bodies.

The experimental method followed by Auerbach was much the same as that of Fromme, except that the core was left in the magnetizing spiral during the make and break of the current. The core was generally a hollow cylinder of soft iron 1484 mm. long, 17*8 mm. in diameter, 1 ’ G mm. thick, with end plates 1 ’ 5 mm. thick. He distinguishes two kinds of magnetic after-effect. The first kind consists in alteration of the magnetization of the body during the action of a constant force, or after it has ceased to act. The second kind is that already mentioned, in which the action of any force is influenced by preceding forces. It is this second kind of after-effect that is dealt with in the paper from which we are quoting.

The leading peculiarity of his view of the phenomenon is the introduction of the force zero, both as a preceding and as a final force. The fundamental principle laid down is the following :—

When the force p, which, following immediately after the force 0, would produce a magnetization T o , is preceded by a series of forces P 1 , P 2 . . . .P n , the magnetization which results is T, differing from T o by an amount N called the after-effect. N is wholly determined by the first of the preceding forces P 2 , which is such that all the forces that act between P 2 and p lie in magnitude between P 2 and p.

This general law is, however, subject to exceptions. For example, let the whole series of forces acting be P 10 , ρ, P 9 , ρ (evidently an extreme case), then experience shows that neither T 10 nor T 9 is the resulting magnetization, but something intermediate, much nearer to T 10 , however, than to T 9 . In order to obtain T 9 a force P o <∕> must be interposed before P 9 ; even then the magnetization varies a little with P o , but, if the stationary condition for P 9 , p be established by alternating P 9 , p many times after applying P o , thus P 10 , p, P o , P 9 , P, P 9 , P, P 9 , P, · the limit is found to be independent of P o , and is held to be the true value of T 8 ∙

In this way, for a given p, T can be determined as a function of P. It is necessary, however, to attend to the following principle,—that, of two preceding forces lying in magnitude on different sides of p, the second determines the after-effect exclusively only when it differs more from P than the first; in other cases both contribute to the aftereffect; in no case does the first exclusively determine the after-effect. In the case where both preceding forces lie on the same side of p, the exceptions to the general law are far less marked; only where the second force is very nearly equal to p does it exercise a disturbing influence on the after-effect of the first.

The process used for obtaining T as a function of P, for a given p, say 10, is therefore to cause the influencing forces to alternate with the influenced, the succession of the former being such that the one preceding p always differs less from p than the one following. The stationary condition is supposed to be established for each pair as above explained; e.g., starting with P=11, the series might be 11, 10, 8, 10, 13, 10, 6, 10, 15, 10, 4, 10, &c. In this way T 11 , T 8 , T 13 , <fcc., can be determined.

When the values of T are plotted against the values of P, the curves corresponding to different values of p have all a similar character (see figure 37). They consist of two congruent parts lying on the two sides of a point of inflexion, which is the only point that has any marked character. To the right of the inflexion the concavity is towards the axis a of P, to the left in the opposite direction. The infinite branches appear to approach asymptotes parallel to the axis of P. The abscissa oi the point of inflexion for any particular curve p is P —p; the ordinate is T p , which may be called the normal magnetization corresponding to p when p alone has acted before. This of course is an ideal case; but a process is indicated for determining T p directly. The dotted curve in the figure is the curve of normal magnetization, whose abscissa and ordinate are p and T p .

From the symmetry of the curves representing the aftereffect Auerbach concludes that the after-effect of forces on opposite sides of p as to magnitude, and equidifferent from it, is equal and opposite, and ascribes the failure to observe the after-effect of forces smaller than p to the interposition of the force zero. He further concludes that the after-effect depends in the same way on P -p as T p depends on p.

There is one of the curves of after-effect, that, viz., for p — ο, which has a special meaning. It is clearly the curve of permanent or residual magnetism, which is thus in Auerbach’s view a particular case of after-effect. To it we can apply the general rule given above, subject of course to like exceptions.

It would be premature to pronounce any opinion as to the ultimate value of Auerbach’s results; but the elegance of his representation of the phenomena will scarcely be disputed. In the latter part of his paper he applies his views to explain the peculiarities in the curve of magnetization with forces of ascending magnitude obtained when the after-effect is neglected, and to the cyclical process discussed by Warburg. He also discusses the influence of the duration of the impulse of the magnetizing force and of the sudden closing and opening of the current. His conclusions agree in the main with those of Fromme: in particular he inclines to Fromme’s view that there is a specific magnetic effect produced in certain cases by the breaking of the current while the core is in the spiral. This effect in certain cases (with short thick cores) is so great that a permanent magnetization of opposite sign to the total induced magnetism remains. This “anomalous magnetization” was first observed by Von Waltenhofen, who also establishes the more general result, of which this is an extreme case, viz., that the residual magnetism of the core depends upon the rapidity with which the magnetizing force is reduced to zero. Auerbach lays down as a general principle that when the variation of the magnetizing force is slow and continuous the velocity of the transition does not influence the final magnetization; but sudden transition causes the final magnetization to be less or greater than that obtained by gradual transition, according as the passage is from a greater to a less or from a less to a greater force.

The reader who wishes to pursue the present subject farther should consult the works of the following experimenters :—

Jamin, who holds what he apparently regards as a new theory of magnetization. It is in point of fact merely a modification of the theory of solenoids, somewhat restricted in its application to the phenomena of magnetic induction. His special point is that the lines of magnetization in a bar magnetized (say) by a magnetizing spiral only penetrate to a limited depth, which is greater the greater the current. The following experiments are adduced in confirmation of his views. The steel tube of a Chassepot rifle was plugged at both ends by screwing into it bolts of the same metal. Inside was placed a cylindrical rod. It was found that, so long as the current in the spiral was not very great, the rod was not sensibly magnetized; but, as the current increased, it became more and more affected, and by and by was as much permanently magnetized as if the enveloping tube had been absent. Again, the rod having been magnetized to saturation and inserted in the tube, a demagnetizing force was applied to the whole, and it was found possible to render the tube and core together seemingly neutral, or even oppositely magnetic, while the rod when taken out proved to be still powerfully magnetized in the original direction. Again, a bar was magnetized by a powerful current, and then magnetized in tho opposite direction by another current. The surface of the bar was then eaten away to a certain depth; and it was found that the original magnetization reappeared. These experiments, although most interesting in themselves, do not appear to warrant the interpretation which their author puts upon them. Jamin has made extensive researches on the magnetic distribution in bars and ribbons of steel, partly with a view to obtain empirical rules for the construction of powerful permanent magnets, in which he has been very successful.

Gaugain, Comptes Rendus, passim; Ann, d. Chim. et d. Phys., (5) xi.

Christiansen, “Researches on the Magnetic Distribution in an Iron Bar, on one part of which is placed a Short Magnetizing Spiral,” Wied. Beibl., i., 1877.

Ruths, “Ueber den Magnetismus weicher Eisencylinder und verschieden harter Stahlsorten” (Dortmund, 1876), Wied. Beibl., i., 1877.

Whipple, “Induction Constants of Permanent Magnets of various shapes, from the determination at Kew,” Proc. Roy. Soc. Lond., 1877.

Oberbeck, “Ueber die Fortpflanzung der magnetischen Induction im weichen Eisen” (Halle, 1878), Wied. Bcibl., ii., 1878.

Külp, “Experimentaluntersuchungen über magnetische Coër-citivkraft,” Carl. Rep., 1880.

Baur, “Experiments with an Iron Ring on the Magnetization Function for very small Forces,” Wied. Ann., xi., 1880.

Riecke, “On the Experimental Test of Poisson’s Theory,” Wied. Ann., xiii. p. 485, 1881.

Siemens, a very interesting paper, “On the Effect of the Magnetization of Iron in any Direction upon its Permeability in the Perpendicular Direction,”· Wied. Ann., xiv., 1881.

Righi, ‘ ‘ Contributions to the Theory of the Magnetization of Steel,” Mem. d. Ace. d. Bologna, 1880; Wied. Beibl., v., 1882.

For a succinct account of several of the foregoing memoirs, see the “Nachträge” to Wiedemann’s Galvanismus, and a paper by the same author in Poggendorffs Annalen, clvii. p. 257, 1876.

Influence of the Hardness and Structure of Iron and Steel on Permanent Magnetism.— Some information has already been given incidentally on this subject, and a lengthy discussion would be out of place here. The statements of the various authorities are very contradictory. This is not to be wondered at; for those best qualified to prepare the materials for experiment are generally deficient in the scientific knowledge requisite to enable them to form a sound judgment as to the result, while thoroughly trained scientific men have not as a rule acquired a command over the delicate manipulation of the forging and tempering of steel, an art which those who possess it usually find difficult to describe in words or reduce to rules. There is the further circumstance that many who have been successful in making good steel for magnetic or other purposes have found it for their interest not to publish the process by which success was attained.

Fineness of grain and uniformity of temper are the greatest requisites in steel for permanent magnets. The latter in bars of any size is never attained in perfection, for the surface is always harder than the interior. The mischief which thereby arises may be understood by taking the extreme case of a thin steel tube magnetized to saturation, and then fitted with a perfectly soft iron core. It is clear that the core will act very much like the armature of a horse-shoe magnet; the lines of force will run back through it, and the external action will be in a great measure destroyed.

The different tempers of steel may be roughly classified as glass hard, straw colour, blue, and soft. The current statement is that the harder the steel the more difficult it is to magnetize, but the better it retains its magnetism. If this were so, provided sufficient magnetizing force to produce saturation were at command, the best temper for magnets would be glass hard. Lamont, however, whose experience was great, states that he found the loss after magnetization to be as great, and to continue as long, with glass hard as with blue tempered magnets. The same experimenter gives it as his opinion that great differences in the quality of magnets arise more from defects as to homogeneity, continuity, and uniformity of temper than from the quality of the steel in other respects; he inclines, however, to a preference for English cast steel.

Purity and homogeneity of structure are equally necessary in iron of high magnetic inductive susceptibility and small coercive force. Hammering, rolling, and drawing diminish the susceptibility and increase the coercive force. Rolling does so more in the direction of rolling than transversely, so that the iron becomes aeolotropic. It is advisable in all cases where high susceptibility is wished to anneal the body carefully after manufacture, by heating it in a wood fire and allowing it to cool very gradually; this process is still more effective when the iron is covered all over beforehand with half an inch or so of clay.

The reader who wishes for further details on this subject should consult Lamont’s Handbuch des Magnetismus, chap. v. The following references to the literature may be useful.

Michell, Treatise of Artificial Magnets, 1750; Coulomb, Mem. del ’ Acad., 178i ; Barlow, Phil. Trans. , 1822; Kater, PhH. Trans., 1821; Sabine, Phil. Trans., 1843; Hansteen, Pogg. Ann., 1825; Hacker, Pogg. Ann., 1848; Poggendorff, lb., 1850; Müller, Ib., 1852; Matthiessen, Phil. Mag., 1858; Airy, Ib., 1863; Von Waltenhofen, Pogg. Ann., 1864; Trêve, Comptes Rendus, 1869; A. I.. Holz, JVied. Ann., v., 1878; Ruths, Wied. Beibl. i. 1877; Chees-man, Wied. Ann. 1882.

Special Magnetic Character of Nickel and Cobatt.— Besides the results of Rowland above quoted, we have on record experiments by the following physicists:—Biot, Traité de Phys., 1806; Gay Lussac, Ann. d. Chim. et d. Phys., 1824; Lampadius, Schwegger's Jour., 1814; E. Becquerel, Comptes Rendus, 1845; Plücker, Pogg. Ann., 1854; Arndtsen, jlb., 1858; Hankel, Wied. Ann., 1877; Becquerel, Ann. d. Chim. et d. Phys., 1879; Gaiffe, Comptes Rendus, 1881; Wild, Wied. Beibl., 1877.

Experiments with Finely Divided Magnetic Metals and with Electrolytic Iron.— These have been made by various physicists, mostly to test the theory of molecular magnets. The earliest of the experiments with finely divided iron was made by Coulomb, who mixed iron filings with wax, and found that the magnetic moment was proportional to the mass of magnetic metal. Similar experiments were made by the elder Becquerel, his result being that the magnetic moment was proportional to the weight of magnetic substance, so long as the filings were not too densely distributed; with increasing density the mixture acquires magnetic properties more like those of a continuous metallic mass. Several modern experimenters have gone into the matter with considerable care; but their results are not sufficiently concordant, or of sufficient general interest, to justify us in dwelling at length upon them here. A few references to recent memoirs will suffice.

Boenιstein, Pogg. Ann., 1875; Toepier and Von Ettingshausen, Ib., 1877; Von Waltenhofen, Wied. Ann., 1879; Auerbach, Ib., 1880; Baur, lb., 1880.

Experiments on electrolytically deposited iron have been made by Beez, Pogg. Ami., I860; Jacobi, Ib., 1873; Beez, Ib., 1874; Holz, Ib., 1875; Baur, Wied. Ann., 1880.

Using a fine scratch on a varnished silver wire as electrode, Beez deposited a thread of iron between the poles of an electromagnet, and thus obtained a permanent magnet of extreme tenuity. It was found that the inductive susceptibility of this linear magnet was very small, and that considerable magnetizing force produced no increase of its permanent magnetism. Thus in one case the original magnetism was 360, the total magnetism under the inducing force 370, the magnetism remaining after the force ceased to act 360. Broader, but equally thin, magnets deposited in a strong field in the same way gave more temporary magnetism than the linear magnets, but never more permanent magnetism than they possessed originally. Thicker plates exhibited greater temporary magnetism, and also an increase of the permanent magnetism acquired during deposition. With continued reversals of the magnetizing force electrolytic iron gave a continual decrease of the temporary magnetism down to a certain limit (as does steel); but the negative permanent magnetism never approaches so near the positive after many reversals as in the case of steel. On the other hand, Jacobi found that iron reduced electrolytically from ferrous sulphate and sulphate of magnesia, even after· tempering, took a considerable temporary moment, but retained very little permanent magnetisfm Holz found that the iron reduced from the solution of Jacobi and Klein was not sensibly hardened by heating and suddenly cooling, although its density was increased, and that its coercive force was diminished. On the other hand, it was found that hard tempering decreased the density of steel. He draws the conclusion that the coercive force is greater the farther apart the molecules. Baur’s main result is that the maximum of magnetization with electrolytic iron occurs for much larger forces than with ordinary iron. These results are not wholly concordant; but the discrepancies may be reasonably assigned to differences in the preparation of the metal.

Magnetic Properties of Matter in General.

Among the earliest statements of the properties of the loadstone we find accounts of its action on other bodies; but it is clear from their surroundings that these statements are purely fabulous. Many experimenters at a later date found indications of magnetic action In other metals besides iron; but with praiseworthy caution they ascribed them for the most part to the admixture of small quantities of iron. There can be no doubt that the results of Cavallo obtained with brass (especially hammered brass) were due to impurity, for Bennet failed to obtain any indications of magnetism with pieces of brass made from pure zinc and copper, whereas he was immediately successful on adding small traces of iron to the metal.

It very soon appeared, however, that an independent magnetic property must be ascribed to nickel and cobalt, and to these were by and by added with more or less certainty manganese and chromium.

Brugmans seems to have been the first to observe the repulsion by a magnet of a body not permanently magnetized. He found that a piece of bismuth floating upon mercury in a small paper boat was repelled by both poles of a magnet. Lebaillif confirmed the observation of Brugmans, and found that antimony possessed a like property. Saigey, who experimented on the same subject, concluded that all bodies when suspended in air behave like bismuth, unless they contain traces of iron.

Notwithstanding these results and others which we pass over, the whole matter remained in obscurity till the repulsion of neutral bodies was rediscovered by Faraday in 1845. He speedily unravelled the laws of the phenomenon, showing how much depends on the nature of the body, and how much upon the nature of the magnetic field. His observations enabled him in fact to comprehend under a few general principles the action of all magnetic bodies whether of the nature of iron or of the nature of bismuth. The earlier observers had fallen into difficulties by neglecting the effects due to heterogeneity of field; these were pointed out for the first time by Faraday, and since then order reigns where there was formerly confusion.

The best arrangement for testing the behaviour of weakly magnetic bodies is to suspend either a small sphere of the substance or else a small cylinder in a heterogeneous magnetic field. This field is usually produced by placing two pointed soft iron poles (fig. 38) on the arms of a powerful electromagnet. The line joining these poles is called the axial direction of the field; directions perpendicular to this line are called equatorial. The magnetic force varies along the axial line, being less in the middle than at the poles; and it decreases everywhere from the axial line outwards. For some purposes poles of the shape shown in figure 39 are used; here the line along the upper edges of the poles are lines of greatest force, whereas the line in the plane of the upper faces equidistant from the upper edges is a line of weakest force; the force also decreases to the right of ab and to the left of cd.

In suspending small spheres the best plan is to hang them from one end of an arm of wood db (fig. 40). At the other end of this arm is placed a counterpoise b, and the whole is suspended by a fibre of unspun silk u from a torsion head t, by means of which the arm db can be brought into any required position, and if necessary kept there by the exertion of a known torsional couple. The arm and suspension must be carefully guarded from draughts by enclosing it in a glass case, which fits over the poles of the electromagnet, and is provided with a door and with means for bringing the torsion head t over any given part of the magnetic field. When a cylindrical piece is to be tested it is suspended from the fibre %soas to hang horizontally. For this purpose Faraday was in the habit of using a stirrup of carefully selected writing paper attached to the lower end of the fibre. It is of the utmost importance to guard against magnetic action on the suspension · the least trace of iron in the arm db for instance, or in the paper stirrup, would in many cases be more than sufficient to mask the action proper to a weakly magnetic body.

In every experiment the magnetic behaviour of the support should be tested by itself beforehand, so that if any residual effect be present it may be allowed for. The greatest caution is also requisite in choosing the material to be experimented upon. There must be no chemical impurity, especially no trace of iron; the spheres and cylinders must not be worked with iron tools or even with dirty hands. A source of error to be specially guarded against in experiments with metals, or other good conductors, is the action arising from induced currents in the mass of the tested body caused by the increase and decrease of the strength of the magnetic field when the circuit of the electromagnet is made and broken. This error is wholly avoided by waiting till the suspended body has come to rest, and attending only to deflexions which are permanent after the intensity of the field has become steady.

The first substance with which Faraday experimented was a bar of the heavy glass with which he had discovered the rotation of the plane of polarization of light. It took up the equatorial position between the poles of the electromagnet as soon as the current was established. There was no distinction between its ends, or according to the direction of the lines of force; the bar always took the shortest course to the equatorial position, and remained there in stable equilibrium. When placed in the axial position it was in unstable equilibrium, and on the slightest displacement either way it moved off in that direction to the equatorial position. A further action was observed when the bar was placed with its centre of mass out of the centre of the field; it was then repelled as a whole away from the nearest pole (no matter which). On testing a small cube or sphere of the substance, no pointing tendency was observed, but the mass as a whole when it was placed unsymmetrically with respect to the poles tended to pass away from the poles towards the centre of the field, and from the axial line outwards.

Faraday sums up the matter by saying that every element of the heavy glass tends to move from places of stronger to places of weaker resultant magnetic force. This is exactly the opposite of the law for bodies like iron (see mathematical theory above, p. 247). All bodies that follow the same law as heavy glass he calls diamagnetics, all that follow the opposite law, like iron, paramagnetics. For the purposes of experimental demonstration it is better to take some weaker paramagnetic than iron, e.g., a tube filled with a solution of ferric chloride; for the order of magnitude of the effect obtained is then the same as with diamagnetics, and there is no danger of complications arising from the mutual action of the particles of the substance (see above, p. 245).

Faraday found the following substances to be diamagnetic; i.e., pieces of them tended to set their longest dimension equatorial between pointed poles, and spheres and cubes of them tended to pass from places of stronger to places of weaker force :—rock crystal, sulphate of lime, sulphate of baryta, sulphate of soda, sulphate of potash, sulphate of magnesia, alum, muriate of ammonia, chloride of lead, chloride of sodium, nitrate of potash, nitrate of lead, carbonate of soda, Iceland spar, acetate of lead, tartrate of potash and antimony, tartrate of potash and soda, tartaric acid, citric acid, water, alcohol, ether, nitric acid, sulphuric acid, muriatic acid, solutions of various alkaline and earthy salts, glass, litharge, white arsenic, iodine, phosphorus, sulphur, resin, spermaceti, caffeine, cinchonia, margaric acid, wax from shellac, sealing wax, olive oil, oil of turpentine, jet, caoutchouc, sugar, starch, gum arabic, wood, ivory, mutton (dried), beef (dried), blood (dried or fresh), leather, apple, bread.

In testing liquids Faraday used a very thin glass tube of the form shown in figure 41; the opening being very fine, there was no need for a cork or other stopper which might have caused disturbance; the slight diamagnetic effect arising from the glass was allowed for. Another way of testing a liquid is to place it in the bottom of a watch glass which rests on the edges of the pole of the electromagnet. When the fluid is paramagnetic, it collects in the places of greater force, forming a depression in the centre of the field as in figure 42; when it is diamagnetic, it collects in the places of weaker force in the centre of the field, as in fig. 43. Yet another method is to put a small quantity of the fluid in a narrow tube, and place the tube horizontally in the equatorial line so that the end of the liquid column is just on the axial line. When the electromagnet is excited the liquid will be driven away from the axial line or drawn in according as it is diamagnetic or paramagnetic.

Faraday found that breaking a weakly magnetic body into pieces, or even reducing it to powder, produced no effect upon its magnetic behaviour provided its general form was unaltered. In order to avoid disturbance from the magnecrystallic effect to be described presently, it is often advisable to reduce certain substances to powder before testing them; the powder is filled into a thin glass tube and then tested like a liquid. By means of powdered bismuth the tendency of a diamagnetic to pass from places of stronger to places of weaker force can be very prettily shown. If the powder be strewn upon the circular end of the core of an electromagnet, it will leave the edges and collect in the centre, whereas iron filings will leave the centre and arrange themselves round the edges, the fact being that at the edges the force is much more intense than in the centre.

Faraday arranges the metals in the following order of descending magnetic susceptibility :— [table]

Silicium is given as strongly paramagnetic, and beryllium, aluminium, potassium, and sodium as weakly magnetic; the last three were given as diamagnetic by Faraday; the magnetic character appears to depend on the method by which the material is prepared, being doubtless determined by the presence or absence of slight impurities. The copper of commerce is magnetic, owing to traces of iron; but when it is reduced by means of zinc from the chloride or sulphate it is diamagnetic. It would appear that the paramagnetism of titanium, palladium, platinum, and osmium is due to iron impurity. Platinum reduced from very pure chloroplatinate of ammonium by heating in a current of air is diamagnetic. According to Graham the magnetism of palladium when charged with hydrogen is due to the presence of hydrogenium; Blondlot, however, has recently found that palladium is less magnetic when charged with hydrogen than when uncharged, from which he concludes that condensed hydrogen is pretty powerfully diamagnetic. Tellurium, sulphur, selenium, and thallium are strongly, and niobium and tantalum weakly diamagnetic.

Magnetic Properties of Gases.— The earliest results of Faraday were of a negative description, but the discovery by Bancalari of the powerful diamagnetic action of flame again drew the attention of Faraday, Pliicker, and Becquerel to the subject. Faraday caused the gas under examination to stream vertically upwards or downwards (according as it was lighter or heavier than the surrounding gas) between the poles of an electromagnet, and observed how the stream was deflected. In the case of colourless gases the deflexion was observed by allowing small traces of hydrochloric acid to mix with the gas, and then placing in different parts of the field small tubes containing pieces of filter paper moistened with ammonia; by noticing in which of these the white fumes of ammonium chloride were formed the course of the gaseous current could be determined. Another method employed was to fix two thin glass tubes containing gases to be tested to the ends of a cross piece on one end of the arm of a torsion balance; the tube containing the most magnetic then moved towards the axial line. Another method, employed both by Plücker and by Faraday, is to blow soap bubbles with the gas to be tested, and observe their behaviour in the magnetic field, allowing of course for the feeble diamagnetism of the water film. Faraday’s results are as follows :— [table]

+ means magnetic relatively to the surrounding gas, — diamagnetic; the epithets strong and weak relate of course to the apparent behaviour under the circumstances of the experiment.

It appears therefore that oxygen is the most paramagnetic of all the gases; on this account Faraday conceived that it probably played an important part as a cause of terrestrial magnetism. Becquerel has concluded from recent experiments of his own that the specific magnetism of ozone is still greater than that of oxygen. Faraday was able by filling thin glass bulbs with oxygen at different densities to show that the magnetic susceptibility decreased with the density, apparently in simple proportion. Some numbers giving an idea of the magnetic susceptibility of the various weakly magnetic bodies are given below.

In all experiments with gases or fluid media, and indeed in every possible magnetic experiment more or less, it is important to notice that the resulting magnetic action is the difference between the action of the movable body and the action on the surrounding medium. This was first pointed out by Faraday. He prepared three solutions of ferrous sulphate. No. 1 contained 74 grains of the hydrated salt for every ounce of water; No. 2 was formed by diluting one volume of No. 1 with two volumes of water, No. 3 by diluting one volume of No. 1 with fifteen volumes of water. Three glasses g v g 2 , g 3 and three tubes i 1 , i 2 , t 3 were filled with the respective solutions. The glasses were placed in succession between the pointed poles of the electromagnet, and the tubes tested in them with the following result :— [table]

4- means pointed axially; 4- 4- the same with greater decision; — pointed equatorially: 0 was indifferent.

We have here the experimental confirmation of the important theoretical conclusion (see above, p. 248) that any body will behave paramagnetically or diamagnetically according as it is surrounded by a medium less or more magnetic than itself. In cases where the square of the susceptibility may be neglected, it is clear that the resultant action on any body is the difference between the action upon it and the portion of the medium which it displaces. This principle, which is the analogue of the Archimedean law for floating bodies, is of great use in quantitative magnetic experiments. It was exemplified by Pliicker, and extensively applied in magnetic observations by BecquereL Becquerel found, for instance, that the differences between the couples tending to set a small rod of sulphur in water and in air, in magnesium chloride and in air, and in nickel sulphate and in air were very nearly the same as the corresponding differences for a rod of wax.

Very curious qualitative illustrations of differential magnetic action are obtained by scattering drops of alcoholic solution of chloride of iron in olive oil; the drops of chloride collect and displace the olive oil in the places of stronger force. Another form of the same experiment consists in placing a layer of oil of violets over a layer of solution of chloride of iron. When a narrow cell filled in this way is placed equatorially with the interface of the two liquids in the axial line, on exciting the electromagnet the iron solution rises in the equatorial plane forming a disk-shaped mass around the axial line. Notwithstanding these results the general opinion of experimenters seems to be that no separation of the parts of a solution can be effected magnetically once the constituents have been thoroughly mixed. Thus Faraday could obtain no evidence of the concentration of an iron solution near the pole of a magnet, although it was exposed for days together in the magnetic field, and found no separation of the oxygen and nitrogen of atmospheric air, although they differ greatly in their magnetic character.

Plücker endeavoured to show that the air enclosed in a vessel placed between the poles of an electromagnet was rarefied by the magnetic action. Faraday, however, with almost identical experimental arrangements arrived at a negative result.

Elaborate investigations of the magnetism of chemical compounds have been made by G. Wiedemann with a view to connect their magnetic properties with their composition. A full account of these researches will be found in Wiedemann’s Galvanismus, Bd. ii. § 590 sgf r The following are some of the more important of his conclusions as to the effect of composition.

1. The magnetic susceptibility of the dissolved salt by itself is nearly independent of the solvent, being proportional to the concentration.

2. If the magnetic moment m induced by a field of unit intensity in a unit of weight of the salt dissolved in water be called the “specific susceptibility,” and the product μ = Am, where A is the molecular weight of the salt, the “molecular susceptibility,” then the molecular susceptibility of the dissolved salt of the same metal with different acids is approximately the same. The mean molecular susceptibilities for nickelous, cobaltous, ferrous, and manganous salts are as 142: 313: 387: 468.

3. The molecular susceptibility of cobaltous salts stands about midway between the molecular susceptibilités of nickelous and manganous salts; and the ferrous salts stand midway between cobaltous and manganous.

4. The molecular susceptibility of dry salts (combined with water of crystallization) is for the most part nearly the same as their molecular susceptibility in solution.

A similar law holds to a certain extent for insoluble salts freshly precipitated; and generally, with like chemical properties of the metallic molecule, the molecular susceptibility remains the same.

5. Two diamagnetic elements may give a magnetic compound; e.g., copper and bromine, both diamagnetic, give bromide of copper, which is paramagnetic.

6. When two solutions are mixed and the salts exchange their constituents by double decomposition, the specific magnetism of the solutions taken together is unchanged. Whence the conclusion is drawn that the susceptibility of a binary compound is made up by addition of the susceptibilities of its constituents, and that these constituents preserve their susceptibilities unaltered when their constitution or atomic arrangement in composition is unaltered.

Magnecrystallic Action.— In what precedes we supposed the inductively magnetized body, whether paramagnetic or diamagnetic, to be isotropic, and all experiments on its magnetic properties to be conducted in a heterogeneous magnetic field. In a uniform field such a body would be acted upon neither by force of translation nor by rotational couple. The case is otherwise if the body be magnetically aeolotropic. In this case, according to the mathematical theory, (1) the body ought to set in a uniform field so as to place its axis of greatest magnetic permeability (i.e., of greatest paramagnetic and of least diamagnetic susceptibility) parallel to the lines of force, and (2) in a heterogeneous field Faraday’s translational force from places of less to places of greater resultant force in the case of paramagnetic, and from places of greater to places of less in the case of diamagnetic bodies, ought to be greatest when the axis of greatest susceptibility is parallel to the lines of force, least when the axis of least susceptibility is in the same position, and intermediate for other positions of the body.

In observing the first class of phenomena above mentioned, poles with flat faces are placed on the electromagnet. Faraday recommends that the faces should be placed at a distance of about one-third of their breadth. He warns the experimenter, however, that the uniformity with this arrangement is by no means perfect, although in general sufficient. The best arrangement would be to use the magnetic field in the interior of a cylindrical coil of sufficient length were it not for the difficulty of attaining the requisite intensity in this way. In cases where there is any doubt it is well to give the body under examination a spherical or cubical shape, and so eliminate the tendency to set arising from heterogeneity of field.

The first observations of the magnecrystallic couple were made by Plücker, and elaborate investigations of the phenomenon were made by him in conjunction with Beer, in the course of which the magnetic properties of a large number of crystalline bodies were examined. Plücker also detected the magnecrystallic property in a rapidly cooled cylinder of glass. Shortly after Plücker’s first results were published, Faraday discovered the magnecrystallic action of crystallized bismuth. At first, misled no doubt by the language in which Plücker stated the newly discovered facts, he did not recognize the identity of the two phenomena; but on further investigation he was able to class all the observations under a few simple laws, which in the mathematical form given to them by Thomson constitute the theory already given. To the observations of Plücker and Faraday Knoblauch and Tyndall added the important discovery that bodies in which the linear density in one direction is greater than in another, whether as a consequence of compression or of stratification artificial or natural, exhibit magnetic aeolotropy.

It is convenient, following the analogy of physical optics, to divide magnetically aeolotropic bodies into (a) “uniaxal” bodies, ue., those that are symmetrical about one principal axis of magnetic susceptibility, or, in other words, have two of the principal coefficients of magnetic susceptibility equal (κ 2 = κ 3 )j and (δ) “biaxal” bodies, i.e., those that have the three principal susceptibilities unequal.

Class (a) naturally divides itself into those in which the susceptibility parallel to the axis of symmetry is greater and those in which it is less than that in the plane perpendicular to it. The former (where κ 1 >κ 2 ) are said to be positive, the latter (κ 1 <κ 2 ) negative uniaxals. We have also to attend to the distinction which arises according as the mass of the crystal is paramagnetic or diamagnetic (κ 1 and κ 2 both +, or both - ). We have then the following experimental behaviour in uniaxal bodies :— [table]

Faraday found for example that a crystal of bismuth suspended with its plane of smoothest cleavage vertical set with this plane perpendicular to the lines of force, but was indifferent when suspended with this plane horizontal. The axis of magnetic symmetry is therefore perpendicular to the plane of smoothest cleavage, and, since the substance is diamagnetic, it is a negative uniaxal. When suspended in any other way the crystal set so that the magnetic axis rested in a vertical plane through the direction of the lines of force. The difference between the behaviour of magnetic and diamagnetic uniaxals is beautifully illustrated by the behaviour of pure Iceland spar, which is a positive diamagnetic uniaxal, and sets the optic axis, which is the magnetic axis, equatorially; when, however, as sometimes happens, the calcium is partly replaced by iron, its physical properties (optical included) being thereby unchanged, except that the mass of the crystal becomes magnetic, it sets the optic axis axial.

All these cases are of course included in the single rule that the body sets the axis of greatest permeability parallel to the lines of force. Since the setting of weakly magnetic aeolotropic bodies depends merely on the differences between the principal magnetic susceptibilities, it follows that it is independent of the medium in which the body is placed. This was established experimentally by Faraday, who found the magnecrystallic couple exerted on a crystal of bismuth to be the same whether it was surrounded by air, by water, or by a saturated solution of protosulphate of iron.

Pliicker gives the following list of uniaxal aeolotropic bodies :— [table]

The phenomena in the case of biaxal aeolotropics are naturally more complicated; but they are all comprehended in the simple rule given above (page 244) that the axis of greatest paramagnetic or of least diamagnetic resultant susceptibility in the horizontal plane tends to set itself parallel to the lines of force, or, in the words of Faraday, the body tends to set so as to allow the greatest number of lines of magnetic induction to pass through it. In the azimuth just mentioned the body is in stable equilibrium, in the perpendicular azimuth in unstable equilibrium. There are two axes of suspension in the plane of the axes of least and greatest susceptibility, viz., the normals to the circular sections of the ellipsoid κ 1 V + κ 2 y 2 + κ 3 s 2 = constant, such that the body behaves indifferently; these axes were called by Pliicker the “magnetic axes” of the body. If we observe the times of vibration T 1 , T 2 , T 3 of a sphere of the substance, when the axes κ 1 , κ. 2 , κ 3 respectively are vertical, then we have at once by the theory already given

T 1 : T 2 : T 3 :: 1/V κ. i - κ i : l∕Vκ 1 - κ 3 : 1/V κ 1 - κ 2 ; whence 1/T® + 1/T’ = 1/T’, and tanω = T 3 ∕T 1 , ω being the angle between either magnetic axis and the axis of greatest magnetic susceptibility. These results were verified experimentally by Pliicker. Magnecrystallic phenomena of the second hind were looked for by Faraday very early in the history of the subject, but at first he was unsuccessful in detecting them. He seems, however, to have understood and clearly represented to himself in his own way their close connexion with the phenomena of the first kind, for he alluded to the subject more than once, and finally in the twenty-sixth series of his experimental researches, where he explained at length his magnetic theory, he showed that such an effect ought to exist, and actually succeeded in observing it in the case of a crystal of bismuth, which he found to be less repelled from places of stronger to places of weaker force when its axis was parallel to the lines of force than when it was perpendicular to them. He concluded that with Iceland spar the translational force ought to be greatest when the axis is parallel to the lines of force, and least when it is perpendicular to them; but his apparatus was not sufficiently delicate to show the effect.

Unlike the magnecrystallic phenomena of the first kind, those of the second kind depend on the difference between the susceptibilities of the body and of the surrounding medium. Faraday demonstrated this conclusion experimentally by covering crystals of the red prussiate of potash with a thin layer of wax to prevent dissolution, and immersing them in solution of sulphate of iron of various strengths between the pointed poles of an electromagnet. In water the crystal was attracted to places of stronger force in all positions, in concentrated solution of sulphate of iron repelled in all positions, while in a solution of 14 or 15 volumes of the concentrated solution to 6 volumes of water it was attracted when the axis of symmetry was parallel to the lines of force and repelled about as strongly when the axis was perpendicular to them. Here then the crystal actually behaved paramagnetically in one direction and diamagnetically in the other. Similar results can be obtained with Iceland spar in a mixture of alcohol and water.

The prediction of the second class of magnecrystallic phenomena is one of the most extraordinary instances of the theoretic insight which formed so large a part of the genius of Faraday. The laws of the phenomena of the first class might be regarded as merely a skilful classification of observed facts, but the passage therefrom to the second class was a step of the first magnitude; it constitutes in fact the root of the whole matter. To Sir William Thomson belongs the credit of throwing the laws of magnecrystallic action into the appropriate mathematical form, and of showing that they range themselves quite naturally under the theory of Poisson.

Tyndall succeeded, where Faraday had failed, in showing the magnecrystallic phenomenon of the second kind in Iceland spar. It is particularly instructive to compare his results for carbonate of iron and carbonate of lime, both positive uniaxals, but the one magnetic and the other diamagnetic. His results are as follows :— [table]

Hankel measured the repulsion on a cylinder of bismuth placed with its magnetic axis inclined at angles of 15°, 45°, and 75° with the lines of force, and compared his results with the theoretical formula 90 - 7 + 45 , 3 sin 2 ≠; the result was as follows :— [table]

Influence of Crystalline Form, Compression, &c., in producing Magnetic Aeolotropy. — In general the magnetic aeolotropy stands in close relation to the crystalline form, and consequently to a considerable extent also to the optical properties. Thus crystals of the regular system exhibit as a rule no magnecrystallic properties, but there appear to be exceptions in the case of certain pyroelectric crystals such as boracite. Again, crystals that have one crystallographic axis of symmetry are usually magnetically uniaxal, but the optical distinction of positive and negative does not involve the corresponding magnetic distinction, as is shown by the results of Tyndall and Knoblauch with pure Iceland spar and Iceland spar in which part of the calcium is replaced by iron. Crystals that are optically biaxal are as a rule magnetically biaxal, but the magnetic properties cannot be deduced immediately from the optical.

If a small cylinder be made of a paste formed with finely ground bismuth and gum water it will point equatorially in a heterogeneous field, but if the roll be squeezed flat the plate thus formed will point axially, although its length be ten times its breadth, A roll of paste of powdered carbonate of iron, again, will point axially, the plate formed by squeezing it flat equatorially. From these results Tyndall and Knoblauch concluded that, if the arrangement of the particles of any body be such Us to present different degrees of proximity in different directions, then the line of closest proximity, other circumstances being equal, will stand axial if the mass be magnetic, equatorial if the mass be diamagnetic. They constructed parallelepipeds (1 in. × ⅛ in. × ∣ in.), first, by gumming together rectangular slips of sandpaper (1 in. × ∣ in.), secondly, by gumming together squares of the same (∣ in. × ∣ in.). The paper was comparatively indifferent, while the sand by itself was magnetic; and it was found that the first model set its longest dimension axially, while the second set its longest dimension equatorially; i.e., the layers of sand set in both cases axially. Tyndall has observed similar magnecrystallic actions with naturally stratified bodies such as shale, and in fibrous bodies such as wood. He was even able by squeezing plates of bismuth to apparently reverse the magnetic character of the substance; for the compression rendered the plates aeolotropic with an axis perpendicular to their longest dimension, and in consequence they set axially like plates of a paramagnetic substance. A crystal of bismuth compressed in a direction perpendicular to the ordinary magnetic axis, i.e., parallel to the planes of principal cleavage, had its behaviour reversed as to the second class of magnecrystallic effects, the ratio of the repulsions when the crystal was set with its original axis axial and with its original axis equatorial having been changed from 71 :100 to 112 :100. It was also found possible by squeezing a ball of bismuth dough unequally in two perpendicular directions to imitate a biaxal magnetic crystal such as heavy spar. Tyndall and Knoblauch attempt to explain the magnetic phenomena exhibited by crystals proper by means of these results. They assume that the planes of cleavage are directions of closer aggregation, and therefore tend to point axially in magnetic and equatorially in diamagnetic crystals. For example, the first of the above-mentioned sandpaper models would represent magnetic crystals that cleave parallel to their axis, the second magnetic crystals which cleave perpendicular to their axis. If we regard this theory merely as a way of representing the facts of observation, even if we allow it to be sufficient, it is far inferior in simplicity to the theory of Faraday and Thomson, the sufficiency of which is not disputed. Regarded as an attempt to penetrate a little farther into the relation between molecular structure and magnetic properties, it is of great interest and importance, even if we admit that like most other speculations of the kind it leads us but a little way; for the question arises immediately, How does proximity of the molecules increase specific inductive capacity? This last question is all the more difficult to answer that no experiment has ever yet been adduced wherein the effect of the mutual induction of the parts of a diamagnetic or weak paramagnetic body plays an undoubted part. Discussion as to the Existence of Diamagnetic Polarity.— Soon after Faraday’s first discovery of diamagnetism, an animated discussion arose as to the proper way of stating the facts involved in the new phenomenon. Faraday himself inclined in the first instance to put the matter by saying that under the action of an inducing force a diamagnetic body is magnetized in a direction opposite to that of soft iron; at a later period he abandoned this form of statement in favour of what he called the theory of magnetic conduction, which fitted better with his ideas as to the part played by the surrounding medium by means of which magnetic action is transmitted from one body to another. Faraday’s first theory under the name of the theory of diamagnetic polarity was immediately adopted by the Continental physicists, such as Weber, Reich, and Poggendorff, who naturally found it consonant with their favourite views as to action at a distance. It was also supported in England by Tyndall and others. Many experiments were advanced on both sides of the question, and the result was much instructive illustration of the laws of magnetic action. But the controversy settled nothing, because in point of fact there was nothing to settle. Either theory was perfectly sufficient, when properly applied, to represent the phenomena, and each left the question of the ultimate nature of paramagnetic and diamagnetic action where it found it. This ought to have been evident after Thomson had shown that the phenomena were included in a perfectly natural generalizatian of Poisson’s theory, indicated in fact by Poisson himself, and demonstrated that Faraday’s conception of the phenomena was only another method of viewing the facts leading to identical conclusions. Faraday himself seems in the end to have considered that the difference was a matter of phrases. Since Clerk Maxwell’s elaborate mathematical reconstruction of the theories of Faraday this seems to be universally recognized, and the discussion has subsided. For a full account of the various interesting experiments that were made during the controversy the reader may be referred to Wiedemann’s Galvanismus, § 558 sq., and to the reprint of Tyndall’s Papers on Diamagnetism and Magne cry stallic Action, pp. 7 6 sq. Numerical Data respecting the Susceptibility of Weakly Magnetic Bodies.· —The earlier experimenters arrived for the most part at the conclusion that the susceptibility κ of weakly magnetic bodies is constant. Among these may be mentioned Weber, who experimented with bismuth, E. Becquerel, Tyndall, Joule, Reich, and Matteucci, who experimented on various substances by means of the torsion balance; Christie, who worked with bismuth, and Arndtsen, who worked with ferric sulphate and ferric chloride, both using Weber’s diamagnetometer; and Wiedemann, who experimented with solutions of a variety of salts. E. Becquerel, however, in some of his experiments, e.g., with sulphate of nickel, found that κ showed a tendency to decrease for very large values of the magnetizing force; Pliicker, ° who tested a great variety of substances (powdered or in solution) by measuring with a delicate balance the attraction or repulsion exerted upon them by an electromagnet, arrived at a similar conclusion; but the methods of both these experimenters are open to suspicion.

A large number of relative results were obtained by the earlier experimenters, but in some cases the methods employed were not satisfactory, and in others the results so evidently depend on the state of aggregation of the material that they are of little importance. The following tables will give the reader some idea of the relative magnitudes of the susceptibilities of different substances :— [table]

The numbers here denote specific magnetic susceptibility; i.e., equal weights of the substances are compared. [table] [table]

The results marked with an asterisk are taken from Becquerel; the rest are from Faraday. The numbers relate to equal volumes of the substances, and the medium is supposed to be vacuum; so that water in air would be represented by 100.

The numbers of Faraday, Becquerel, and Matteucci agree very fairly; e.g., according to Faraday the susceptibilities of water, oxygen, and air are as - 100: + Γ8: + ’ 352, according to Becquerel as - 100: + l·82: + '382. Pliicker’s results do not agree so well with those of Faraday and Becquerel; but his method was faulty.

Within the last five years a large number of absolute determinations of κ have been made, chiefly for bismuth and ferric chloride. Toepier and Von Ettingshausen in their experiments on bismuth used with some alteration the method of induced currents employed by Weber in the earliest attempts that were made to determine the susceptibility of bismuth. Like Weber they compare bismuth with iron, an unsatisfactory procedure on account of the great variability of the susceptibility of iron for different magnetizing forces, and for different samples with the same magnetizing force.

Silow worked with ferric chloride. In his first set. of experiments he observed the time of vibration of an astatic needle suspended over a cylindrical vessel filled with the solution; in his second investigation the solution was placed in a glass globe, on the outside of which insulated wire was wound so that a given current in it produced a uniform magnetic field whose strength could be calculated; the deflexion of a properly astatized needle suspended inside the globe, was observed when the globe was empty and when it was full, and thence κ was calculated; in his third determination he used the method of Toepier and Von Ettingshausen so improved as to allow an absolute determination of κ to be obtained directly.

Borgmann enclosed one coil within another, and filled the hollow cylindrical space between them with the solution of ferric chloride to be tested; he also used the ring method of Stoletow and Rowland.

Jacques, following a method elaborated by Rowland, measured the repulsion of crystals of bismuth and Iceland spar placed with their magnetic axes axially and equatorially between the poles of a Ruhmkorff’s electromagnet, the field of which was carefully explored after the manner of Verdet by means of a small coil moved through a known distance in different parts of it; from these observations the two principal magnetic susceptibilities were calculated.

Schuhmeister experimented with ferric chloride, using the same method as Rowland and Jacques.

In the experiments of Eaton the method formerly employed by Wiedemann was adopted; the data in his paper are, however, insufficient for an absolute determination either of the magnetizing force or of κ ; in fact he determines merely the force with which the magnetic body is attracted and the magnetic moment of the electromagnet, assuming that the strength of the magnetic field at a given point is proportional to the latter,—which is not necessarily true, for the magnetic distribution in the core of the electromagnet may alter with increasing current.

Von Ettingshausen in the most recent research on the subject with which we are acquainted has made determination of the susceptibility of bismuth by four different methods. The first of these was that formerly used by Toepler and himself, with the addition that the action of the bismuth bar was compared with that of a solenoid of as nearly the same dimensions as possible through which flowed a current of given strength (an artifice previously used by Christie). The second method consisted in measuring the force with which a portion of the diamagnetic substance hung in the axis of a coil and near one of its ends was repelled out of the coil when a known current passed through it. The third method was that of Rowland and Jacques. The fourth consisted in measuring the deflexion of a magnetometer needle produced by placing a piece of the diamagnetic substance between the poles of a powerful magnet under whose action the magnetometer needle had come to rest in the first instance. The agreement between the results obtained by all the different methods was very fair considering the smallness of the effects to be measured in some of them. The second method is pronounced to be the best, and by means of it he gives also a determination of κ for ferric chloride.

Some of the results of the different experimenters for bismuth are given in the following table :— [table]

Most of the specimens contained slight traces of iron. Although the range of the magnetizing force in Von Ettingshausen’s experiments was considerable, κ was very nearly constant; if there was any tendency to variation, it was decrease with the large magnetizing forces.

The results for ferric chloride are not so concordant. Silow, after comparing his own earliest result (10 6 κ = 81 for a solution of density 1 ’ 475, magnetized by the earth’s horizontal force) with those of Borgmann (10 6 κ = 48"8, density 1 ’ 87, magnetic force 40 to 59), concluded that the susceptibility of ferric chloride probably follows the same law as that of iron ; ue., it first increases, then reaches a maximum, and afterwards decreases more or less slowly. His later experiments confirm this conjecture, and he finds that κ has a maximum value for a magnetizing force of about ∙4 C.G.S. The smallest force used was about ’ 08 C.G.S. and the corresponding value of 10 6 κ was 34; the largest value of 10 6 κ occurring in his tables is 179. The values obtained in his last investigation are smaller than those given in his first table, but there is the same increase and decrease. The following are his latest results :— [table]

The unit of is the earth’s horizontal force, presumably at Moscow.

From the observations of Arndtsen on a solution of density 1*495 Silow calculates 10 θ κ = 57 ’ 5 (magnetic force 20 - 3). For a solution of density D395, with magnetizing forces from 38 to 252, Schuhmeister gets 10 e κ=30 to 39. Von Ettingshausen, for a solution of density U48, with magnetizing force 14 to 20, gets 10 6 κ = 59 to 56.

The following are the values of 10 6 κ obtained by Schuhmeister for various substances. [table] [table]

Relation of Magnetism to other Physical Properties. Shocks, Jarring, or Vibration.— The effect of these in aiding the action of an inductive magnetic force was known to Gilbert; and it was also known to the earlier experimenters that the permanent magnetism of a body not subject to external magnetizing force was destroyed by like causes. The action is precisely similar to that found in the case of bodies temporarily or permanently deformed by mechanical stress, and, again, to the first effects of temperature on bodies temporarily or permanently strained, or temporarily or permanently magnetized. The effect may be conceived as consisting of a loosening of the molecules for the moment, so that they follow more easily any force acting on them whether mechanical or magnetic. The following parallel statements, taken from the results of Wiedemann, who has devoted much careful study to these phenomena, will sufficiently illustrate the matter :— [table]

Minuter details regarding the effects of jarring will be found in memoirs by Wiedemann, Fromme, Auerbach, and others already quoted. The reader may also consult Warburg, Pοgg. Ann., 1870, and Villari, Pοgg. Ann., 1869.

Mechanical Strain produced by Magnetization.— The starting point of accurate research on this subject was the discovery made by Joule in 1842 that a bar of soft iron lengthened when it was temporarily magnetized in the longitudinal direction. When the magnetizing force was removed the bar shortened, but in general not quite to its original length. This residual extension was due in part to permanent magnetism, but he found that the permanent magnetization due to a current 1088 was reversed from - 1 3 to + ’ 25 by a current 175, while two-thirds of the permanent extension was still left. The actual elongation of an iron bar magnetized to saturation was found to be from τo ⅛ 7πy th to ^γ∏⅛ rσ th of its whole length. The extension varied approximately as the square of the intensity of magnetization (temporary or permanent). The general character of the phenomena is the same in soft or hard iron, and in soft or hard steel; but the effects are smaller with hard than with soft bars.

It was found that longitudinal compression of the bar influenced the magnetic extension little if at all; on the other hand, longitudinal traction was found to diminish it, and in the case of thin wires under considerable tension the magnetization caused a contraction. Thus in the case of a bar 1 foot long, ∣ inch in diameter, with a weight of 600 lb, there was neither extension nor contraction, even with a current of 1600; with weights of 1040 ft and 1680 B⅛, and a current of 1804, there was a contraction of •00002 inch and *000032 inch respectively. The contraction under tension was found to vary approximately as the product of the magnetizing current and the intensity of magnetization. After the magnetizing force was withdrawn the wire regained its original length, permanent magnetization notwithstanding.

Joule made careful experiments to determine whether the magnetization of an iron bar produced any alteration of its volume, but could find none. He therefore concluded that the longitudinal extension of a magnetized bar is accompanied by an equal lateral contraction; and, in accordance with this conclusion, he found that when an iron tube is circularly magnetized, perpendicular to its length, by passing a current along its axis, it contracts longitudinally.

The results of Joule have been verified and to some extent added to by Wertheim, Buff, Beez, Tyndall, ® Mayer, ® Righi, and Ader. The three first experimented with magnetizing coils shorter than the bar, and found that the extension was much greater when the coil was near the free end than when it was near the fixed end of the bar. This of course raises the question how far the extension is due to electromagnetic action between the coil and the bar, and how far to internal molecular disturbance. Mayer’s results are in agreement with Joule’s except in the case of bars of soft steel, which (not under traction) when the magnetizing current was first established, elongated in some cases ana retracted in others,—at the first break elongated, and subsequently retracted at make and elongated at br’eak. Righi’s results for longitudinal magnetization are in agreement with those of Joule; he also gives a variety of interesting results regarding the effects of circular and longitudinal magnetization on the length of iron wires. Barrett has recently arrived at the interesting result that nickel behaves oppositely to iron,—retracting about τ 8 ¾ 1 g -⅛⅛th when magnetized to saturation; he gives for the elongation of iron and cobalt under like circumstances ⅞< i ⅞ ι ⅞7∏jth and ττ ⅛ o th respectively.

Effect upon Magnetization of Traction along the, Lines of Magnetization.— Matteucci seems to have been the first to discover that when a bar subject to a magnetizing force in the direction of its length is stretched in the same direction its temporary magnetization increases. When the stretching force is removed the magnetization again diminishes. Wertheim confirmed Matteucci’s observation. Villari, however, found that, after the first effect, which is always increase, the application of the traction will cause increase if the intensity of magnetization is not beyond a certain critical value, but decrease if that value is surpassed; the removal of the traction causes in each case the opposite effect to the application.

The effect of the first traction on the permanent magnetization, whether of iron or steel, is a diminution; the effect of subsequent tractions in steel is a diminution on application, with increase on removal; in soft iron an increase on application, a diminution on removal. Partial demagnetization of a steel bar by an opposite magnetic force causes it to behave like soft iron; when the demagnetizing force is sufficient to reverse its polarity, the effect of even the first traction maybe to increase the magnetization.

Sir W. Thomson has carefully studied the phenomena in question, as exhibited in a very soft iron wire '075 cm. in diameter permanently stretched by a weight of 1 lb, and alternately stretched by weights of 7 δ>, 14 δ>, or 21 K>, and unstretched (so that there was no permanent elongation). As the magnetizing force was increased, the increase of magnetization caused by the application of traction increased to a maximum, then diminished, and became zero for a certain critical value of the magnetizing force; after the critical value was passed, the traction caused a diminution of the magnetization, which increased asymptotically towards a fixed limit as the magnetizing force was increased more and more. The following table will give an idea of the results.

I denotes the maximum increase, and D the limit of the decrease, roughly estimated in the same arbitrary unit; is the force corresponding to I, and ‰ the critical force, each expressed in terms of the earth’s vertical force at Glasgow as unit; T is the traction, t the temperature. [table]

Bars of nickel and cobalt were also examined; and it was found that after the first effect the result of applying traction in the direction of magnetization was in both cases to diminish the magnetization. The effect appeared to increase up to a maximum, and then to diminish as the magnetizing force increased; but the critical value was not reached with the largest forces employed.

Traction perpendicular to the lines of magnetization was found by Thomson to diminish the magnetic susceptibility. The experiment was made by means of a gun barrel magnetized longitudinally, and subjected to internal hydrostatic pressure.

.The effect of pressure along or perpendicular to the magnetization would in all probability be opposite (and equal 1) to that of an equal amount of traction; but no experiments have as yet been made on the subject. The effect of traction is therefore to produce magnetic aeolotropy, the susceptibility being increased in the direction of the stress and diminished in the perpendicular direction so long as the intensity of magnetization is not above a certain critical value; above that value the effects are reversed. The effect of pressure would be opposite in every particular. Hence the effect of a shearing stress would be increase βί. magnetic susceptibility along the principal axis of elongation, and decrease (to an equal extent?) along the principal axis of compression.

Delations between Torsion and Magnetization.— These were investigated by Matteucci, and after him by Becquerel and Wertheim. The whole subject was carefully studied by G. W. Wiedemann, who has done more than any living physicist both in discovering new facts in this interesting field and in coordinating those formerly known. We extract from his Galvanismus the following series of parallel statements, which will serve the double purpose of making the reader acquainted with the principal facts, and of drawing his attention to the close analogy between the mechanical and magnetic properties of bodies, and to the almost perfect reciprocity of their experimental laws. [table]

The alterations of the longitudinal and circular magnetization of iron wires may be shown by the induced currents thereby caused in a coil surrounding the wire or in the wire itself. For example, if an iron wire be circularly magnetized by passing a current through it, and then twisted in either direction, an induction current flows through the wire in the same direction as the original current; and an opposite current is observed when the wire is untwisted again. This shows that twisting the wire diminishes the permanent circular magnetization, while untwisting partially restores it.

The relation between bending stress and magnetization has been studied by Guillemin, Wertheim, Ader, and Kimball; but the results are not of sufficient interest to be cited here. The question lias also been raised whether magnetization affects the elasticity of bodies, and has been answered by Wertheim and Wartmann in the negative. Both Kimball 9 and Piazzoli find that the breaking tension of iron wires is increased by longitudinal magnetization; the former puts the increase at 0.9 per cent, when the wire is saturated.

This is the place to mention the so-called ‘-'magnetic sounds” which accompany the magnetization and demagnetization of the strongly magnetic metals. It is now established beyond all doubt that these sounds have their origin partly at least in the mechanical strains accompanying magnetization. In many cases direct magnetic or electromagnetic action, and even electrostatic and thermal actions, concur in producing them, and it is often difficult to say how much is due to each of these several causes. This is especially to be observed where the sounds are produced by the passage of interrupted or undulatυry currents through wires of the strongly magnetic metals. A full discussion of the matter belongs more properly to the subject of electric telephony; but a few notes on the history and literature of the subject may be given here.

Page seems to have been one of the first to notice phenomena of the kind; but Joule appears to have first stated clearly that magnetic-mechanical strain was a specific cause. He says that the magnetic extension in the core of an electromagnet takes place so suddenly that the shock is sensible to the touch, and is accompanied by a musical note arising from vibration in the metal. Marrian, Matteucci, Beatson, and Wertheim all took up the matter; and De la Rive published many investigations concerning it. In 1861 Reiss published the invention of an electric telephone for the transmission of music and speech, which depended essentially on the magnetic sounds produced by a varying current in an iron core. This instrument was the prototype of the telephone of Gray, and of the still more famous instrument of Bell, whose action, although often described as purely electromagnetic, is no doubt in part due to the magnetic strains. Among the more recent investigations on this subject may be mentioned Ferguson, Brοc. Roy. Soc. Belin. , 1878 and 1880; Ader, Comptes Rendus, 1879; Du Moncel, Ib. ; Chrystal, Nature, vol. xxii., 1880; Hughes, Proc. Roy. Soc. Bond., xxxi. and xxxii., 1881.

General Demarhs.— Wiedemann has remarked with justice that most of the effects of strain upon magnetization and vice versa are complex. Apart from the possible admixture of direct magnetic action, we must distinguish (1) the mere disturbing effect of jarring: thus the first application of a mechanical stress has the same effect as a shock, i.e., it loosens the molecules of the body, as it were, and renders them more ready to follow any inductive magnetic force, while the first effect of magnetization upon a body under stress is precisely similar, and may in fact be imitated by mechanical jarring pure and simple; (2) after-effect, whether mechanical or magnetic, the consequence of which is that the effect due to any mechanical stress or magnetizing force is affected by pre-existing stress and magnetization; (3) the proper effect of mechanical stress or magnetic force, which appears at once where one or the other is applied, and disappears when it is removed.

In his excellent analysis of the phenomena, Wiedemann coordinates them throughout by means of an extension of Weber’s theory of “molecular magnets” {Drehbare Mole-cularmagnete). This of course involves an attempt to pass beyond the mere results of experience; and there can be no question that, on the whole, this theory explains the facts in a highly instructive and suggestive manner. The main defect in it is the multitude of assumptions and the want of clearness and definiteness in its conclusions. Thus it is sometimes not easy to see why exactly the opposite conclusion should not be drawn; and it appears hopeless to bring it to the test of a quantitative comparison with experiment.

Without entering into the ultimate causes of magnetism, we might endeavour to reduce the phenomena to the smallest number of experimental facts. Thus, assuming merely the effects of longitudinal and transverse traction upon magnetization and the magnetic extension and compression along and perpendicular to the lines of magnetization, we might explain many of the results concerning the relation between torsion and magnetization.

Let us take for example No. VI. of Wiedemann’s parallel statements. In fig. 44 let the upper end of the wire be the fixed end, and let P be a point in any of the thin coaxal cylindrical shells into which the wire may be supposed divided. First suppose the wire to be circularly magnetized by the action of a downward current, the resultant magnetic force at P being in the horizontal direction PB. If now the wire be twisted in the direction of the arrow T, it acquires two axes of greatest and least magnetic susceptibility Pe and Pr. The resultant magnetic force PB being resolved along these axes will induce more magnetism along Pe than along Pr; hence the aeolotropy will cause the resultant magnetization to take the direction PB'; it will therefore have a positive vertical component downwards, which agrees with statement VI. In fact the twisting converts the circular lines of magnetization into right-handed helices.

Next let us suppose the wire untwisted to begin with, but magnetized both circularly and longitudinally, the components being PB and PA. The resultant magnetization will then have some direction such as Pe, but, by Joule’s principle, this will cause extension along Pe and compression along the perpendicular direction Pr; consequently the wire will twist in the direction of the arrow T, which agrees with statement 6. Moreover, since magnetization along PB alone would simply cause the tube to expand along a horizontal section, and magnetization along PA alone would simply cause longitudinal extension, it is clear that when either A or B is given the twisting reaches a maximum and then diminishes when the other is increased.

It does not seem unreasonable to expect that a general mechanical theory of this kind will yet be found to coordinate all the facts; although there are difficulties in its way at present. The phenomena will then be reduced to two or three experimental facts at the utmost, which it will be the business of some ultimate dynamical theory of magnetism to explain.

Effect of Temperature.— Some information on this subject has been given incidentally above, p. 256. We collect here a few additional facts; but a complete account of all that has been done could not be compressed within our available space, owing to the great diversity of opinion upon the subject. That the question is a very difficult one will appear at once, if we reflect that variations of temperature influence the density and molecular structure of magnetic bodies to a remarkable degree, and that thus secondary influences arise in addition to the proper effect of temperature.

That very high temperatures destroy both the magnetic susceptibility aud the power of retaining magnetism altogether has been known since the infancy of magnetic science. Thus Gilbert found that a loadstone and a piece of iron equally lost their power of affecting the magnetic needle when heated very hot, and remarks that the magnetic property returns to the iron after it has cooled a little, but that the magnetic virtue of the loadstone is altogether destroyed. Similar results were obtained by Brugmans, Boyle, Cavallo, Barlow and Bonnycastle, Christie, Ritchie, Erman, Scoresby, Seebeck, and others. Faraday found that a steel magnet lost its permanent magnetism rather suddenly at a temperature a little under the boiling point of almond oil; it behaved like soft iron till it was raised to an orange-red heat, and then it lost its magnetic susceptibility and became indifferent. The temperature at which retentive power for permanent magnetism was lost appeared to vary in steel with the hardness and structure; in fragments of loadstone it was very high: they retained their permanent magnetism until just below visible ignition in the dark, but, on the other hand, they lost their susceptibility at dull ignition, i.e., at a much lower temperature than iron. Nickel was found to lose its magnetic susceptibility at a much lower temperature than iron, viz., about 330° to 340° C. Cobalt is much more refractory, for it retains its susceptibility, according to Faraday, nearly up to the melting point of copper, i.e., to a white heat. The writer had occasion to verify these results in the course of some experiments on the magnetic sounds in wires of iron, nickel, and cobalt traversed by an interrupted current of electricity.

The effect of extreme cold, produced in the ordinary way by means of solid carbonic acid and ether, was, according to Trowbridge, to diminish the moment of a steel magnet (magnetized at 20° C.) by about 60 per cent.

The effect of moderate alteration of temperature varies greatly according to circumstances. We shall consider separately the effect upon the magnetic susceptibility and upon the permanent magnetism; but it must be noticed that no such separation is possible in actual experiment.

The temporary magnetism of bars of cast iron, smithy iron, soft iron, soft steel, and hard steel magnetized by the earth’s vertical force was found by Scoresby to be insensible at a white heat, but to be much greater at a dark red heat than at the temperature of the air. The difference was most marked in the case of hard steel, no doubt partly because of the softening of the bar. Similar experiments were made by Barlow, Seebeck, and others. Kupfer experimented on the subject using variations of temperature between 0° and 100° C., and found the susceptibility of soft iron to increase with the temperature. Wiedemann’s conclusion is that the first alteration of temperature, whether increase or decrease, increases the temporary magnetism of iron or steel, whatever the temperature at starting. If the temperature be repeatedly altered and brought back to its initial value, the magnetization continues to increase, but after a time becomes more and more nearly constant at the initial temperature. After this state has been reached, an increase of temperature causes increase of magnetization in very hard steel bars, a decrease of temperature a decreàse of magnetization; the behaviour of soft steel bars is exactly opposite.

Baur and Wassmuth have recently taken up the matter with all the advantages of modern experience. The former concludes from his experiments on iron by the ring method, at temperatures between 0° and 150° C., that the magnetic susceptibility for a given magnetizing force increases with the temperature if the force be below a certain critical value (3 - 6 or so), but decreases as the temperature increases if the force be above that value. The smaller the magnetizing^force the greater the influence of temperature on the magnetic susceptibility. The result of his experiments at very high temperatures is that, for small magnetizing forces, the susceptibility at first increases rapidly as the temperature increases, reaches a maximum at red heat, and then falls suddenly to zero. For large forces, the susceptibility decreases gradually until red heat, and then falls suddenly to a very small value. According to him, if a bar be cooled from white heat the first traces of susceptibility are observed at a very bright red, - the brighter the greater the magnetizing force. He gives a variety of interesting results concerning the phenomenon of Gore, all in accordance with what we have just stated.

In his earlier researches Faraday was unsuccessful in obtaining any evidence of the influence of temperature on the susceptibility of weakly magnetic bodies, such as the chlorides of the magnetic metals or of diamagnetic bodies. His earliest results were obtained with gases, and that too, strange to say, before the magnetic character of gases was fully investigated. It was Bancalari’s discovery of the extraordinary behaviour of flame between the poles of an electromagnet that led Faraday to resume his magnetic experiments on gases. Flames of all descriptions are strongly repelled from the axial line of a heterogeneous magnetic field,—so much so that it is impossible to induce the flame of a candle to go between the pointed poles of a powerful electromagnet when they are placed at a short distance apart. The flame is blown aside, or even downwards, as if by a strong current of air issuing from between the poles. If a flat pointed flame is placed with its centre a little below the axial line, when the magnet is excited it drops down and spreads out below and around the axial line, assuming a fish-tail shape. It appears that the effect is not due to the solid matter in the flame but simply to the hot gases in it ; for the upper and cooler part of the stream of smoke from a freshly extinguished taper is scarcely affected, while the lower and hotter part is most powerfully acted upon, being blown aside and often split into two independent streams. A careful investigation led Faraday to the conclusion that oxygen, carbonic acid, and coal gas are rendered more diamagnetic, or, what is the same thing so far as the resultant differential action is concerned, less magnetic by heat, and that this effect was much greater than could be accounted for by the mere rarefaction of the gas. He likewise obtained an increase of the susceptibility of oxygen by cooling it with ether and solid carbonic acid. Nitrogen appeared to be altogether indifferent. He found in a later series of experiments that the magnecrystallic property of bismuth was destroyed at a temperature a little below its melting point, and that the same thing happened to crystalline antimony a little below red heat. In the thirtieth series of his experimental researches he states that between 35° and 142° C. the susceptibility of a specimen of spathic iron ore perpendicular to its magnetic axis decreased by *333 per cent, per degree centigrade of rise of temperature; this agrees very closely with the formula which was found by Wiedemann to represent very approximately the temperature effect for salt solutions, viz., h∙ l = h 0 (1 — Ό0325 t). For the decrease in the magnecrystallic couple, or, which is the same thing, in the difference between the susceptibilities along and perpendicular to the magnetic axis, he found for the spathic iron ore ’ 482 per cent, between 0° and 138° C., and the percentage of (decrease was four times as great between -14° and 0° as between 129° and 143°. The corresponding decrease in the case of crystalline bismuth between 36° and 137° C. was , 53 per cent. The experiments of Pliicker and Matteucci led them to conclude that the susceptibility of diamagnetics diminishes with increase of temperature; in the case of bismuth the decrease between ordinary temperatures and its melting point is said to be about one-sixth or more.

Canton seems to have been one of the first to study the effect of moderate variations of temperature on the permanent magnetism of iron and steel. The results of his and Hallström’s experiments went to show that permanent magnetization decreases when the temperature rises, and increases again when the temperature falls. In reality, however, as was shown by Kupffer, Riess and Moser, G. Wiedemann, and others, the phenomenon is complicated; for, if we repeatedly heat a magnet and allow it to cool to its initial temperature, the magnetization lost at each heating is only partially recovered on cooling, and thus a progressive ι loss goes on, until at last a constant state is reached, in which the magnetization lost on heating is completely recovered on cooling. In this respect, as well as in the effect on the magnetic susceptibility already discussed, there is an analogy between the effect of temperature and the effect of strain; i.e., there is a first or permanent effect and a proper or temporary temperature •effect. The permanent effect is that any alteration of temperature, be it increase or decrease, diminishes the permanent magnetization just as a shock or a jar would do, and probably for a similar reason. The proper or temporary effect consists in a decrease of magnetization with increase of temperature, which is completely recovered on decrease of temperature and vice versa. If this be borne in mind, together with what has already been said above, it will not be difficult for the reader to see that the order and amount of the temperature variations, the hardness and form of the bar, and its magnetic history will all influence the temperature coefficient.

To take one example, Wiedemann found that a bar magnetized at 0° C. and then partially demagnetized by an opposite force, lost magnetism when heated; if the demagnetization was not carried too far, it did not when cooled again to 0° wholly recover what it had lost. If the demagnetization was carried a certain length, it recovered all that it had lost; if farther still, more than it had lost. It was in fact found possible to demagnetize a bar, so as to render it apparently unmagnetic, and then to restore part of its original magnetism by merely heating and cooling it again. Similar phenomena were observed with a bar magnetized and demagnetized at 100°, and then alternately cooled and heated. Unverdorben, who arrived somewhat later at similar results, represents the matter by saying that the bar in this case has two magnetizations superposed, each having its own temperature coefficient.

The following are a few additional references to sources of information concerning the present subject: Mauritius, Pogg. Ann., 1863, and Phil. Mαg., 1864; Jamin and Gaugain, Comptes Rendus, passim·, Favé, Ib., 1876; Poloni, Wied. Beibl., 1878.

Development of Heat during Magnetization.— Reasoning on purely thermodynamic principles from the results of Faraday, as to the influence of temperature on the magnetic properties of bodies, Thomson has concluded—(1) that a piece of soft iron at a moderate or low red heat, when drawn gently away from a magnet, experiences a cooling effect, and, when allowed to approach, a heating effect, and that nickel at ordinary temperatures and cobalt at high temperatures (between the melting point of copper and some lower temperature) experience the same kind of effect; (2) that cobalt at ordinary temperatures and up to the temperature of maximum permeability experiences a cooling effect when allowed to approach a magnet, and heating when drawn away; (3) that a crystal in a magnetic field experiences cooling when the axis of greatest paramagnetic or of least diamagnetic susceptibility is turned from along to across the lines of force, and vice versa.

Besides these considerations, the fact that those who adopt the molecular magnet theory are obliged to assume something of the nature of a frictional resistance to the turning of the magnetic molecules, and generally, without reference to any particular theory, many of the phenomena of coercive force, lead us to suppose that some specific development of heat may accompany magnetization and demagnetization. The experimental verification of this suspicion is, however, a matter of great difficulty, owing to the enormous generation of heat arising secondarily from induced currents in the mass of the metal. The development caused by magnetization and demagnetization was taken advantage of by Joule in one of his determinations of the mechanical equivalent of heat, but he makes no attempt to separate the effect of the two causes, indeed it did not concern his purpose to do so. Notwithstanding that several experimenters have attacked the problem, it cannot be said that it is yet completely solved. It will therefore be best simply to call the reader’s attention to some of the papers that have been published on the subject, and leave him to form his own judgment.

See Von Breda, Pοgg. Ann., 1846; Grove, Phil. Mαg., 1849; Edlund, Pοgg. Ann., 1864; Villari, N. Cirn., 1870; Cazin, Comptes Rendus, 1874; Herwig, Wied. Ann., iv., 1878; Trowbridge, Wied. Beibl., 1879.

Miscellaneous Relations of Magnetism to other Physical Properties.— According to Maggi the thermal conductivity of magnetized iron is less along the lines of force than across them. Naccari and Bellati were unable to verify this result; Tomlinson, however, found, that the conductivity of iron and steel bars was diminished by longitudinal and increased by transversal magnetization.

Abraham, Edlund, Mousson, and Wartmann all made experiments in search of a magnetic alteration of the electric conductivity of iron. Thomson seems, however, to have been the first to arrive at any definite result. He found the conductivity to be diminished along the lines of magnetization and increased across them. Beez verified the former result, but doubts the latter, which he is inclined to explain as a secondary effect caused by the compression of the iron arising from the external magnetic action on the plates used in Thomson’s experiments.

Thomson also found that the thermoelectric quality of iron was affected by magnetization; the thermoelectric current flowed from unmagnetized to longitudinally magnetized, and from transversely magnetized to unmagnetized or longitudinally magnetized iron through the hot junction. In the case of nickel, the current flowed from longitudinally magnetized to unmagnetized through the hot junction, i.e., nickel behaved oppositely to iron. Thomson’s results have been in part confirmed by a recent investigation of Strouhal and Barus.

A relation between magnetism and light was first established by Faraday’s discovery of the magnetic rotation of the plane of polarization of a ray passing along the lines of force. This subject belongs more properly to physical optics, but there is one magnetic phenomenon apparently closely connected with it which falls to be mentioned here. This is Hall’s discovery, ix. and x., 1880. ] that, if an electric current flow in a thin metallic strip in a direction AB, the effect of placing the strip in a magnetic field with its plane perpendicular to the lines of force is to cause a transverse electromotive force perpendicular to AB, which changes in sign when the direction either of the current or of the magnetic field is changed. This transverse electromotive force is proportional to the product of the current intensity and the strength of the magnetic field; caeteris parilnιs, its direction in the case of iron is opposite to that in other metals, and its magnitude is also greatest with iron. This discovery establishes the existence of the rotatory coefficient of resistance mentioned by Maxwell in his discussion of aeolotrdpic conductivity; and Rowlaud has shown that the phenomenon is probably due to the same cause as the magnetic rotation of the plane of polarization., x., 1880, and xι,, 1881. ]

If, as modern physicists suppose, magnetism be a dynamical phenomenon, time must enter as a conditioning element. The question has been raised how long any magnetizing force takes to develop the maximum magnetization that it is capable of producing. There are many facts that go to prove that this time is very small, or, at all events, that any force develops a very large fraction of the total magnetization due to it in a very short period of time. Perhaps the most wonderful evidence on this head is the fact that the telephone, which depends essentially on varying magnetic action, can reproduce the sounds of human speech even to the consonants., ii. 1876. ] Experiments bearing directly on the subject have been made by Villari. A flat circular disk of flint glass was placed between the poles of a Ruhmkorff’s apparatus for measuring the magnetic rotation of the plane of polarization. The axis of the disk was perpendicular to the axial line, so that rotation brought the different radii successively into the line of sight. When the disk was at rest the magnetic action in one experiment caused a rotation of 19 divisions; spinning the disk at the rate of 110, 121, 143, and 180 turns per second reduced the magnetic rotation of the plane of polarization by 2, 5, 10, and 17 divisions respectively; the reduction was less the greater the magnetic force. From this Villari concluded that in flint glass not less than 0Ό01244 second is required to produce such a diamagnetic intensity as can be observed by the rotation of the plane of polarization, and that 0Ό0241 second at least is required to develop the greatest diamagnetization of which this substance is capable; he also states that the diamagnetism lasts for less than ΟΌ0018 second after the inducing force is withdrawn. A series of interesting experiments on the oscillation of the plane of polarization caused by the oscillatory discharge from a Leyden jar recently made by Bichat and Blondlot led them to a different conclusion, viz., that if any lagging of the induced magnetization behind the magnetizing force exists it is less than ΟΌ00033 second. No explanation has been given of the discrepancy of these results.

In the early part of this century there was an animated controversy as to whether light exerted a direct influence upon magnetization, in which Morichini, Mrs Somerville, Christie, Riess and Moser, and many others took part. Nothing definite, however, was established. A similar fate befell the attempts to trace the influence of magnetic force upon crystallization, and to detect a relation between magnetism and gravity, although both quests at one time or another engaged the skill of Faraday.

Forms, Construction, and Preservation of Magnets.

This subject occupied a large portion of most of the earlier treatises on magnetism. Much of the information given, however, either has now been recognized to be of questionable value or has been superseded by recent progress, and retains a merely antiquarian interest; a few brief remarks, mainly historical, will therefore be sufficient.

The oldest form of magnet was a piece of magnetic iron ore or loadstone. The power of these natural magnets varied exceedingly from one specimen to another. An elaborate discussion of the various kinds of loadstone will be found in Gilbert’s De Magnete. In order to increase the carrying power, the loadstone was usually fitted with armatures of soft iron upon its polar regions; figure 45, taken from Gilbert, represents one of the oldest arrangements.

Figure 46 is taken from a loadstone in the collection of physical apparatus belonging to the university of Edinburgh, the carrying power of which is 205 ft>. A loadstone in the Teylerian Museum at Haarlem has a carrying power of 230 lb; and one at Lisbon, pre sented by the emperor of China to King John V. of Portugal, is said to support as much as 300 ft>. Small loadstones are often very powerful in proportion to their weight; e.g., Newton is said to have worn in a ring one that weighed only 3 grains, and yet was able to carry about 746 grains; and one in the physical collection at Edinburgh, formerly belonging to Sir John Leslie, weighing itself 3⅛ grains, had at one time a carrying power of 1560 grains.

The introduction of steel magnets, and the perfection to which they were gradually brought, caused the loadstone to fall into disuse. It is said that Galileo possessed the art of making steel magnets about the beginning of the 17th century. It was early discovered that the earth’s force could be utilized in magnetizing steel.

Gilbert was aware that a feeble magnetization could be produced in this way; and Michell, in his treatise on artificial magnets, minutely describes how weak magnets may be made by means of the earth’s force, then combined into bundles or “magazines” and used in turn to produce stronger magnets, these used to produce still stronger, and so on.

The earliest process of all was no doubt the method of rubbing or touching by another magnet. This method of making magnets was studied with much attention by the natural philosophers of the 18th century, among whom we may mention Savery, Knight, Duhamel, Le Maire, Canton, Michell, Aepinus, Coulomb, and Euler. The method of single touch consists simply in stroking the bar to be magnetized alternately on its two halves with the south and north poles of a loadstone or bar magnet, the stroke beginning always at the middle and ending at the end. According to Lamont, the best plan is to lay the magnet flat, overlapping one half of the bar to be magnetized, and then draw it off; when the magnet is held perpendicular to the bar during the process, the result is apt to give an irregular magnetization: e.g. , we may even get a magnet with its two ends north poles and with a south pole in the middle, or one with four poles, a north and south pole at the two ends and a south and north pole in the middle.

The first improvement on single touch was double touch with separate magnets. This consists in using two magnets simultaneously on the two halves of the bar undergoing magnetization. The north pole of one and the south pole of the other are placed either close together, or at a small distance apart near the middle of the bar, and then each is drawn towards the end of the half on which it lies; according to Lamont, here, as in single touch, the magnets should be laid flat on the bar. Michell introduced the further improvement of using two bar magnets (or bundles of such) fastened together and kept parallel at a small distance apart by means of small pieces of wood, the north pole of one being conterminous with the south pole of the other. This pair is placed vertical with one end on the middle of the bar, drawn towards one end and slipped off, then replaced on the middle and drawn to the other end, and so on alternately until the moment of the bar ceases to increase any farther. Instead of the pair of bar magnets a horseshoe magnet might of course be used.

Le Maire introduced the essential improvement of placing the bar to be magnetized upon a larger bar, and then magnetizing the two together. The advantage of this is best seen in the form of the same device adopted by Canton and Duhamel, who magnetized steel bars in pairs, connecting them up parallel to each other by means of two pieces of soft iron, and then magnetizing them in opposite directions. It is easy to see that the magnetization of the one reacts on the magnetization of the other and strengthens it. Michell 7 obtained a similar advantage by magnetizing a number of bars placed end to end in a line; he found, as was to be expected, that the end bars were weaker, but this defect he remedied by repeating the process with the bars arranged in a different order. Coulomb’s method was to place the ends of the bar on the north and south poles of two bar magnets arranged in line at the proper distance apart. This process of connecting up the bars to be magnetized in a closed magnetic circuit is sometimes called circular touch; it can be applied to horse-shoe magnets by placing a pair of them with their ends together, and then passing round and round upon them a horse-shoe magnet or a pair of bar magnets arranged as already described.

Immediately after (Ersted’s discovery of the magnetic action of the galvanic current, Arago, Boisgiraud, ° and Davy almost simultaneously applied this property to the magnetization of iron and steel. Powerful electromagnets, with cores of soft iron, were first constructed a few years later by Sturgeon and Brewster. Pohl, Moll, and Pfaff in Germany, and Henry and Ten Eyck in America, may be mentioned as the most successful of the early constructors. One of the electromagnets of Henry and Ten Eyck reached a carrying power of 2061 ft; but magnets specially constructed for carrying power have surpassed this limit. As a specimen of scientific toys of this description may be mentioned the electromagnet of Roberts (fig. 47), which consists of a square block of iron deeply slotted with four parallel grooves into which three layers of copper wire cable are wound in zigzag fashion so that the current converts the flanges alternately into north and south poles; the armature is a square block planed to fit the face of the magnet. The carrying power of a machine of this kind was 2949 ft, i.e., more than 1⅛ tons!

The forms of electromagnet used in the arts, e.g., in electric bells, fire alarms, telegraphs, telephones, electric light regulators, dynamo machines, &c., are simply innumerable. It will be sufficient to allude to those constructed for the purpose of producing an intense magnetic field, uniform or non-uniform, over a larger or smaller area; these find their practical application in the construction of dynamo-electric machines, but they are mainly interesting to purely scientific men on account of their use in the investigation of the properties of weakly magnetic bodies. Figure 40 shows the usual arrangement adopted for large laboratory magnets. In considering the greatest available strength of such magnets, it is necessary to bear in mind the fact that magnetic saturation of iron is practically reached with magnetic forces much under the greatest that we can command. The strength of field in a narrow crevasse perpendicular to the lines of magnetization in saturated iron is less than 18,000 C.G.S. units; and this is practically the utmost at present attainable, for any addition to the strength of the field, arising from direct action of the magnetizing helix, would not under ordinary circumstances affect the hundreds in this number. Further increase of magnetizing current after we have reached λvithin a small percentage of the limit of saturation is a waste of power.

Elias of Haarlem seems to have been the first who applied the electric current directly with success in the manufacture of powerful permanent magnets. He used a short flat magnetizing coil which was pushed backwards and forwards along the bar, the ends of which were caused to abut against two pieces of iron, which becoming inductively magnetized reacted on the bar, and also served to keep the magnetization at the ends more uniform. The famous Logeman magnets were constructed by this process.

By far the most convenient way of magnetizing steel is to use an electromagnet. The bar to be magnetized may be laid flat on the pole of the magnet before it is excited, and after excitation drawn slowly off. By repeating this process several times, with the north pole of the electromagnet for one half and the south pole for the other half, saturation can be very quickly obtained. Perhaps a better plan is to lay the bar with its ends on the two poles, and then excite the electromagnet. For reasons already sufficiently explained, it is advisable to hammer the bar with a mallet while the magnetizing force is in action, and to turn the current off and on several times in succession.

On account of the difficulty of tempering steel to any great depth from the surface, and for specific magnetic reasons as well, it has been customary in constructing powerful permanent magnets to build them up of thin laminae of steel, each of which is separately magnetized. Figure 48 represents an arrangement of this nature adopted by Coulomb, and figure 49 a horse-shoe magnet constructed in the same way. It will be observed that the ends of the laminae are not exactly conterminous, the middle ones projecting more than the others; this arrangement was adopted with the view of getting rid to some extent of the weakening effect which the induction of one lamina has upon the other. That such an effect exists and is very great was conclusively shown by Coulomb; how far the modification in question cures it is another matter; much no doubt depends on the purpose for which the magnet is required; but it is scarcely worth while to discuss the subject here. We may call attention to a farther point in the construction of Coulomb’s magnet, viz., that the ends of the laminae are embedded in two soft iron terminals N and S; there can be no doubt that, for some purposes at least, this is an advantageous arrangement. Among the famous modern makers of permanent magnets Hacker of Nuremberg, Logeman and Wetteren of Haarlem, Willward, and Jamin deserve to be specially mentioned.

In the preservation of permanent magnets it is essential to avoid extreme changes of temperature and shocks. When the magnet is laid aside it should be made part of a closed magnetic circuit; in the case of a horse-shoe magnet this is attained by simply laying a piece of soft iron, called the keeper, across the poles; bar magnets should be kept in parallel pairs, north pole to south pole and south pole to north pole, with two pieces of soft iron between the poles. When this is done the induced magnetism reacts on the magnets and diminishes the demagnetizing force; the action of shocks then ceases to destroy the permanent magnetism, and may even increase it.

Ultimate Theories of Magnetic Phenomena.

If we pass over the stream theory, which, although partially developed by Euler, has never taken root in modern physical science, the first great theory that we find proposed with a view to the explanation of magnetism is the two-fluid theory of Coulomb and Poisson. This is not an ultimate theory in the modern sense, inasmuch as it is not dynamical; but it was, doubtless, looked upon as ultimate in the days when the imponderable fluids had a recognized role in the physical sciences. In the two-fluid theory the imaginary positive and negative attractive agents (called magnetism in the empirical theory developed above) are regarded as imponderable fluids; but the essential point in the definite form of the theory due to Poisson is that he regards a body susceptible to magnetic induction as made up of an infinite number of particles of infinite permeability immersed in an impermeable medium. After pointing out that, if the particles were of elongated form, and arranged so that the axes of elongation had one preponderating direction, or if they were arranged so that the linear density in different directions varied, the result would be aeolotropy, he assumes that they are spheres uniformily distributed in the impermeable medium so that the volume of the magnetic particles in unit volume of the substance is the fraction h. The problem of magnetic induction under the influence of a uniform force is then the same as the problem of electric induction for an infinite number of perfectly conducting spheres uniformly distributed in a non-conducting medium. He finds for the permeability w =(1 + 2½)∕(1-4

Maxwell has pointed out one fundamental objection to this theory, viz., that the value of k calculated from the formula just given by means of observed values of w in the case of iron is greater than it would be even if the magnetic spheres were packed in the closest possible manner. Another objection is that the theory affords no explanation of the variability of k with different forces. We might of course modify the hypothesis, as was done by Plücker, by supposing that a resistance depending on the magnitude of the force opposes the separation of the fluids in the magnetic molecules, and that in certain cases a frictional resistance tends to prevent their reunion. We might in this way explain magnetic saturation and permanent magnetism; but the theory thus burdened has no more scientific value than the purely empiric theory, and, moreover, affords no clueto the phenomena of diamagnetism.

Iu a very important class of modern theories, the fundamental assumption is that the molecules, or at all events a certain proportion of the molecules, of magnetic substances are small permanent magnets. In a body which is to outward appearance unmagnetized, the axes of these molecular magnets are turned indifferently in all directions; in a body which is magnetized in a certain direction a larger proportion than usual of the molecular magnets have their axes more or less in that direction. Magnetic induction is supposed to consist, not in any alteration of the molecular magnets themselves, but in the orientation of their axes under the action of the inducing force. The reader may figure to himself the nature of the action by imagining a line of small magnetic needles with their axes all horizontal, but all pointing in different directions; the whole system thus arranged will have no determinate magnetic moment, and will represent an unmagnetized body. Next, suppose a magnetizing force to act parallel to the line joining the centres of the needles, they will then arrange themselves in that line, and the magnetic moment of the system will be the sum of the moment of the different parts; we have thus an image of a body magnetized by induction.

The notion of molecular magnets seems to have been suggested by Kirwan; but it was not until a definite form was given to it by Weber that it acquired any importance. The mathematical problem presented is one of great complexity. In the position of equilibrium any molecule is acted on by the magnetizing force, by a magnetic force due to the combined action of the other molecules, and possibly by a force arising from the displacement as well. Weber assumes that the couple tending to restore the molecule to its original position is that due to a constant magnetic force D, parallel to the original direction of its axis. If m be the magnetic moment of a molecule, and there be n molecules in a unit of volume, then the magnetic intensity $ due to the magnetizing force is given by ⅛==2rnn^∕3D, if ⅛j<D; and by $ = rnn(l - D 2 ∕3¾) 2 ), if

> D. In other words, the curve (∣, H) is straight till it reaches the point ( ∣mn, D), it then becomes concave towards the axis of ⅛j, and rises towards an asymptote parallel to the axis of ∣); the maximum value of $ is mn. The theory does, therefore, give a general explanation of the phenomena of magnetic induction. The reader will be able by comparison with the experimental data given above to see how far it falls short of a complete explanation.

If the magnetic substance be devoid of coercive force, we must suppose that the molecules return to their original positions when the magnetizing force is removed. In substances capable of being permanently magnetized, i we must imagine something of the nature of a frictional resistance to the motion of the magnetic molecules; so that, when they are deflected through more than a certain angle, they retain a permanent set after removal of the magnetizing force. Maxwell has worked out the particular hypothesis that each molecule which is deflected through an angle less than β 0 returns when the magnetizing force ceases to act, but that a molecule deflected through an angle β>β 0 retains the deflexion β-β 0 ∙ Denoting D sin β 0 by I., he finds as the result of the above supposition that the curve of temporary magnetization is a straight line from Ü = 0 to $) = I.; after that it is concave to the axis of and rises to an asymptote, the maximum value of $ being mn as before. The curve of residual magnetization begins when = I.; it is concave to the axis of $, and rises to an asymptote corresponding to the maximum ⅛' = jmn{l + √l-I. 2 ∕D s } 2 . It results from the hypothesis that, when a bar is permanently magnetized by a positive force ^j 1 , its magnetism cannot be increased by a positive force < ^ 1 , but may be diminished by a negative force <‰ and, when the bar is exactly demagnetized by a negative force $ 2 , it cannot be magnetized in the opposite direction without the application of a force > but a positive force < H 2 is sufficient to begin to remagnetize the bar in the original direction.

Behind the molecular magnet theory there arises the question, What is the nature of the magnetic molecule? One answer to this question is given by the hypothesis of Ampère, that around each such molecule a current circulates in planes perpendicular to the axis of the molecule. That such an arrangement will be equivalent to an infinitely small magnet in the axis of the molecule, so far as external action is concerned, we know from the laws of electrodynamics. It remains only to inquire what the nature and properties of these molecular currents must be, to trace the full logical consequences of the assumption, and to compare them with experience. This was first done by Weber, and afterwards more completely by Clerk Maxwell.

It is obvious in the first place that the circuits in which the molecular currents flow must be perfectly conducting; for otherwise the electrokinetic energy of the molecular currents would be continually transformed into heat, and a constant supply of energy from without would be necessary to support the magnetism of a permanent magnet, which is contrary to experience. Let A be the effective area of a molecular circuit, I. its coefficient of selfinduction, θ the inclination of its axis to the inducing force 7 0 the primitive current, and 7 the current after the inducing force is in action. Then γ=-=7 0 -∣jAcos0∕I.; and the component of the moment parallel to $ will be A(7 o -^Acos0∕I.)cos0. There are three different cases to consider.

1. Let either 7 0 be so great, or ^A∕I. be so small, that the effect due to the electromagnetic induction may be neglected in comparison with the effect due to the deflexion of the molecule; putting m=A7 0 , we have thus merely the theory of molecular magnets already explained.

2. Let the force resisting the turning of the molecules be infinitely great, we then find for the magnetic susceptibility the value κ= - ⅜nA 2 ∕I.. This is the theory originally proposed by Weber to explain diamagnetism.

3. If the effects due to deflexion of the molecules and to electromagnetic induction in the molecular circuits be both considered, we have a theory intermediate to (1) and (2), inclining to the one or the other according to the assumptions made as to the relative values of 7 0 , A, and I..

The reader will find a full discussion of the different cases in Maxwell’s Electricity and Magnetism, vol. ii. chap. xxii.

The most important attempt that has yet been made to realize a mechanism affording a dynamical explanation of magnetic phenomena is the theory of molecular vortices, published by Clerk Maxwell in the Philosophical Magazine for 1861 and 1862 (4th ser., vols. 21 and 23). The general results, stripped of all particular assumptions, will be found embodied in his great treatise on Electricity and Magnetism; but the following summary, taken from the original paper, may be of some interest.

1. Magnetoelectric phenomena are due to the existence of matter under certain conditions of motion or of pressure in every part of the magnetic field. The substance producing these effects may be a certain part of ordinary matter, or it may be an aethcr associated with matter.

2. The condition of any part of the field through which lines of magnetic force pass is one of unequal pressure in different directions, the pressure being least along the lines of force, so that they may be considered as lines of tension.

3. This inequality of pressure is due to vortices coaxial with the lines of force. The density of the revolving matter is proportional to the magnetic permeability of the medium. The direction of rotation is related to the direction of the line of force; and the velocity at the circumference of the vortex is proportional to the resultant magnetic force.

4. The vortices are separated from each other by a single layer of round particles; so that a system of cells is formed, the partitions being layers of these particles, and the substance of each cell being capable of rotating as a vortex.

5. The particles forming the layer are in rolling contact with both the vortices which they separate, but do not rub against each other. They are perfectly free to roll between the vortices and so to change their place, provided they keep within one complete molecule of the substance; but in passing from one molecule to another they experience resistance and generate irregular motions which constitute heat. These particles play the part of electricity. Their motion of translation constitutes an electric current; their rotation serves to transmit the motion of the vortices from one part of the field to another; the tangential pressures thus called into play constitute electromotive force; and the elastic yielding of the connecting particles constitutes electric displacement.

Maxwell deduces without difficulty all the principal electrical and magnetic phenomena from this theory; and he points out that its general conclusions have a value which does not depend upon the somewhat intricate kinematical arrangements supposed to exist in the magnetic medium. The theory certainly affords us a most instructive dynamical picture of the phenomena of electricity and magnetism; and it remains, so far as we know, the only successful attempt of its kind. (g. ch.)